Multidimensional scaling

Trevor F. Cox, M.A.A. Cox1584880945, 9781584880943

The theory of multidimensional scaling arose and grew within the field of the behavioral sciences and now covers several statistical techniques that are widely used in many disciplines. Intended for readers of varying backgrounds, this book comprehensively covers the area while serving as an introduction to the mathematical ideas behind the various techniques of multidimensional scaling.

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Table of contents :
CRC Press Downloads and Updates……Page 0
Multidimensional Scaling……Page 4
Contents……Page 6
Preface……Page 11
1.1 Introduction……Page 12
Ordinal scale……Page 14
Number of ways……Page 15
1.2.2 Multidimensional scaling models……Page 16
Unidimensional scaling……Page 17
Unfolding……Page 18
1.3 Proximities……Page 19
1.3.1 Similarity/ dissimilarity coefficients for mixed data……Page 25
I.3.3 Similarity of species populations……Page 29
I.3.5 The metric nature of dissimilarities……Page 32
1.3.6 Dissimilarity of variables……Page 33
Example……Page 35
Nominal and ordinal data……Page 23
1.4 Matrix results……Page 36
1.4.2 The singular value decomposition……Page 37
An example……Page 38
Generalized SVD……Page 39
1.4.3 The Moore-Penrose inverse……Page 40
2.2 Classical scaling……Page 41
2.2.1 Recovery of coordinates……Page 42
2.2.2 Dissimilarities as Euclidean distances……Page 44
2.2.3 Classical scaling in practice……Page 46
2.2.5 A practical algorithm for classical scaling……Page 48
2.2.6 A grave example……Page 49
2.2.7 Classical scaling and principal components……Page 53
Optimal transformations of the variables……Page 54
2.2.8 The additive constant problem……Page 55
2.4 Metric least squares scaling……Page 59
Least squares scaling of the skulls……Page 61
2.5 Critchley’s intermediate method……Page 62
2.6 Unidimensional scaling……Page 63
2.6.1 A classic example……Page 65
2.7 Grouped dissimilarities……Page 67
2.8 Inverse scaling……Page 68
3.1 Introduction……Page 71
A simple example……Page 72
3.1.1 Rp space and the Minkowski metric……Page 73
3.2 Kruskal’s approach……Page 74
3.2.1 Minimising S with respect to the disparities……Page 75
3.2.2 A configuration with minimum stress……Page 78
3.2.3 Kruskal’s iterative technique……Page 79
3.2.4 Nonmetric scaling of breakfast cereals……Page 81
3.2.5 STRESS l/2, monotonicity, ties and missing data……Page 83
3.3 The Guttman approach……Page 85
Differentiability of stress……Page 86
Limits for stress……Page 87
3.4.1 Interpretation of stress……Page 89
3.5 How many dimensions?……Page 98
3.6 Starting configurations……Page 99
3.7 Interesting axes in the configuration……Page 100
4.1 Other formulations of MDS……Page 103
4.2 MDS Diagnostics……Page 104
Robust parameter estimation……Page 106
4.4 Interactive MDS……Page 108
4.5 Dynamic MDS……Page 109
An example……Page 111
4.6 Constrained MDS……Page 113
4.6.1 Spherical MDS……Page 115
4.7 Statistical inference for MDS……Page 117
Asymptotic confidence regions……Page 120
4.8 Asymmetric dissimilarities……Page 126
5.1 Introduction……Page 132
5.2 Procrustes analysis……Page 133
Optimal rotation……Page 135
5.2.1 Procrustes analysis in practice……Page 136
5.2.2 The projection case……Page 138
5.3 Historic maps……Page 139
5.4.1 Weighted Procrustes rotation……Page 141
5.4.2 Generalized Procrustes analpsis……Page 144
5.4.3 The coefficient of congruence……Page 146
5.4.4 Oblique Procrustes problem……Page 147
5.4.5 Perturbation analysis……Page 148
6.2 Monkeys……Page 150
6.3 Whisky……Page 152
6.4 Aeroplanes……Page 155
6.5 Yoghurts……Page 157
6.6 Bees……Page 158
7.2 The classic biplot……Page 161
7.2.1 An example……Page 162
7.2.2 Principal component biplots……Page 165
7.3 Another approach……Page 167
7.4 Categorical variables……Page 170
8.1 Introduction……Page 172
8.2 Nonmetric unidimensional unfolding……Page 173
8.3 Nonmetric multidimensional unfolding……Page 176
8.4 Metric multidimensional unfolding……Page 180
8.4.1 The rating of nations……Page 184
9.2 Analysis of two- way contingency tables……Page 187
9.2.1 Distances between rows (columns) in a contingency table……Page 190
9.3 The theory of correspondence analysis……Page 191
9.3.1 The cancer example……Page 193
A single plot……Page 196
9.3.2 Inertia……Page 197
9.4.1 Algorithm for solution……Page 199
9.4.2 An example: the Munsingen data……Page 200
9.4.3 The whisky data……Page 201
9.4.4 The correspondence analysis connection……Page 203
9.4.5 Two-way weighted dissimilarity coefficients……Page 204
9.5 Multiple correspondence analysis……Page 206
9.5.1 A three-way example……Page 208
10.2 The Tucker-Messick model……Page 210
10.3.1 The algorithm for solution……Page 211
10.3.2 Identifying groundwater populations……Page 213
10.3.3 Extended INDSCAL models……Page 215
Carroll-Chang decomposition of W i……Page 216
10.5 PINDIS……Page 217
11.1.1 The theory……Page 221
Level constraints……Page 222
The optimal scaling phase……Page 223
Model estimation phase……Page 224
11.2 SMACOF……Page 225
11.2. I The majorization algorithm……Page 226
11.2.2 The majorizing method for nonmetric MDS……Page 229
11.3 Gifi……Page 230
11.3.1 Homogeneity……Page 231
HOMALS……Page 232
12.1 CANDECOMP, PARAFAC and CANDELINC……Page 237
12.2 DEDICOM and GIPSCAL……Page 239
12.3 The Tucker models……Page 240
12.4 One-mode, n-way models……Page 242
12.5 Two-mode, three-way asymmetric scaling……Page 247
12.6 Three-way unfolding……Page 249
References……Page 250
A.l Computer programs……Page 272
Minimum system requirements……Page 273
DOS Users……Page 274
A.2.3 To run the menu……Page 275
Data manipulation programs……Page 276
The data sets……Page 277
Figures in the text……Page 278
Example 1: Classical scaling of the skull data……Page 280
Example 2: Nonmetric MDS of the Kellog data……Page 281
Example 4: Individual differences scaling of groundwater samples……Page 282
Example 6: Reciprocal Averaging of the Munsingen data……Page 283
A.5.1 Data format……Page 284
Dissimilarities for Individual Differences Scaling……Page 285
Indicator Matrix……Page 286
DAT2UNF……Page 287
LINEAR……Page 288
MDSCAL-T……Page 289
NONLIN……Page 290
RECAVDIS……Page 291
UNFOLDIN……Page 292
VECJOIN……Page 293
VEC-PLOT……Page 294