Procrustes Problems (Oxford Statistical Science Series)

Contents......Page 8
Preface......Page 14
Acknowledgements......Page 15
1 Introduction......Page 16
1.1 Historical review......Page 17
1.2 Current trends......Page 19
1.3 Overview of the topics in the book......Page 22
1.5 Notation......Page 24
2 Initial transformations......Page 26
2.1 Data-scaling of variables......Page 27
2.1.1 Canonical analysis as a form of data-scaling......Page 28
2.1.3 Distance matrices......Page 29
2.2.1 Differing numbers of variables......Page 30
2.3 Types of data sets......Page 31
3.1.1 Least-Squares criteria......Page 35
3.1.2 Correlational and inner-product criteria......Page 36
3.1.4 Quantifications......Page 37
3.2 Translation......Page 38
3.2.1 Simple translation......Page 39
3.3 Isotropic scaling......Page 40
3.5 A note on algorithms......Page 41
3.6 A K-sets problem with two-sets solutions......Page 42
4 Orthogonal Procrustes problems......Page 44
4.1 Solution of the orthogonal Procrustes problem......Page 45
4.2 Necessary and sufficient conditions for optimality......Page 46
4.4 Example of an orthogonal rotation of two sets, including a scaling factor......Page 47
4.5 Different dimensionalities in X[sub(1)] and X[sub(2)]......Page 48
4.6.1 Rotations and reflections......Page 49
4.6.2 The two-dimensional case......Page 51
4.6.4 Best plane rotation......Page 53
4.7 Best rank-R fit, principal components analysis and the Eckart忘oung theorem......Page 56
4.8.1 Orthogonal Procrustes by maximising congruence......Page 57
4.8.2 Robust orthogonal Procrustes......Page 58
5 Projection Procrustes problems......Page 63
5.2 Necessary but not sufficient conditions......Page 64
5.3 Two-sided projection Procrustes by inner-product......Page 65
5.3.1 Tucker\'s problem......Page 66
5.3.2 Meredith\'s problem......Page 67
5.3.3 Green\'s problem......Page 70
5.4 Projection Procrustes by least squares......Page 71
5.4.1 The Kochat and Swayne approach......Page 73
5.4.2 The wind model......Page 75
5.6 Maximising the projected average configuration......Page 80
5.6.2 Method 2......Page 81
5.8 Some geometrical considerations......Page 84
6 Oblique Procrustes problems......Page 88
6.1 The projection method......Page 91
6.2 The parallel axes or vector-sums method......Page 93
6.3 The cases T = C[sup(-1)] and T = (C\')[sup(-1)]......Page 95
6.4 Summary of results......Page 98
7.1 Permutations......Page 99
7.2 Reduced rank regression......Page 100
7.3 Miscellaneous choices of T......Page 101
7.5 Double Procrustes problems......Page 102
7.5.1 Double Procrustes for symmetric matrices (orthogonal case)......Page 103
7.5.2 Double Procrustes for rectangular matrices (orthogonal case)......Page 104
8.1.1 Translation with weighting......Page 106
8.1.2 General forms of weighting......Page 109
8.2 Missing values......Page 111
8.3.1 Pre-scaling R......Page 112
8.3.3 Estimation of T with post-scaling S......Page 113
8.3.6 Row scaling......Page 115
9 Generalised Procrustes problems......Page 118
9.1.1 Generalised Procrustes criteria and some basic identities......Page 119
9.1.2 Algorithms......Page 121
9.1.3 Missing values without scaling......Page 123
9.1.4 Generalised Procrustes analysis with isotropic scaling......Page 127
9.1.5 Contribution of the kth set to the residual sum-of-square......Page 129
9.1.6 The constraint (9.20) and other possibilities......Page 131
9.1.7 Missing values with scaling......Page 133
9.1.8 Exhibiting the group average......Page 134
9.2 Special transformations......Page 136
9.2.1 Generalised orthogonal Procrustes analysis......Page 137
9.2.2 Example of a typical application of GPA in food science......Page 139
9.2.3 Generalised projection Procrustes analysis......Page 141
9.3.1 Example of generalised pairwise Procrustes analysis......Page 144
10.1 Basic formulae......Page 147
10.2 Geometrical unification of orthogonal and projection Procrustes methods......Page 148
10.2.1 Analysis of variance of configurations......Page 149
10.2.2 Partitioning the analysis of variance into components representing projections into independent sub-spaces......Page 152
10.2.4 An example of ANOVA......Page 155
10.3 Criteria as terms in the ANOVA......Page 156
10.4 Discussion......Page 157
11 Incorporating information on variables......Page 161
11.1.1 Linear biplots......Page 162
11.1.2 Biplots in multidimensional scaling of transformed dissimilarities......Page 164
11.1.3 Biplots in multidimensional scaling of transformed variables......Page 165
11.2 An example: coffee images......Page 166
12.1 Introduction......Page 171
12.2 Probability distributions of Procrustes loss values......Page 172
12.3.1 Random data method......Page 174
12.3.2 Random permutation method......Page 176
12.4 Dimensionality of results......Page 177
13.1 Variant treatments of generalised Procrustes analysis......Page 179
13.1.1 Isotropic scaling of the group average......Page 180
13.1.2 Anisotropic scaling of the group average......Page 181
13.1.3 Procrustean individual difference scaling......Page 184
13.2 Individual differences scaling......Page 186
13.2.1 Example of INDSCAL......Page 188
13.3 SMACOF etc.......Page 189
13.4 Structuration des Tableaux A Trois Indices de la Statisitique......Page 190
13.5 Links with canonical correlation......Page 191
13.6 Common components analysis......Page 196
13.7 Other methods......Page 197
14.1.1 Personality psychology/factor analysis......Page 198
14.1.2 Sensory analysis......Page 199
14.1.4 Shape analysis......Page 200
14.1.5 Biometric identification and warping......Page 204
14.1.7 Image analysis......Page 205
14.2.1 Comparing criteria......Page 206
14.2.2 Comparing algorithms......Page 207
14.3 A note on software......Page 208
14.4 Conclusion......Page 209
Appendix A: Configurations......Page 210
B.1 Rotations......Page 212
B.2 Some special cases......Page 214
B.3 General orthogonal matrices......Page 215
Appendix C: Orthogonal projections......Page 218
C.1 Orthonormal operators as rotations......Page 219
Appendix D: Oblique axes......Page 220
Appendix E: A minimisation problem......Page 222
E.1 Zero values of z[sub(i)]......Page 224
E.2 Some special cases......Page 227
E.3 Some remarks on the case k = 0......Page 229
E.4 Note on an alternative parameterisation......Page 230
Appendix F: Symmetric matrix products......Page 231
F.1 Special case......Page 233
References......Page 235
C......Page 244
H......Page 245
N......Page 246
T......Page 247
Z......Page 248