Procrustes Problems

, by Gower, John C.; Dijksterhuis, Garmt B

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ISBN: 9780198510581 | 0198510586
Cover: Hardcover
Copyright: 4/8/2004

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Procrustean methods are used to transform one set of data to represent another set of data as closely as possible. The name derives from the Greek myth where Procrustes invited passers-by in for a pleasant meal and a night's rest on a magical bed that would exactly fit any guest. He theneither stretched the guest on the rack or cut off their legs to make them fit perfectly into the bed. Theseus turned the tables on Procrustes, fatally adjusting him to fit his own bed.This text, the first monograph on Procrustes methods, unifies several strands in the literature and contains much new material. It focuses on matching two or more configurations by using orthogonal, projection and oblique axes transformations. Group-average summaries play an important part and linkswith other group-average methods are discussed. This is the latest in the well-established and authoritative Oxford Statistical Science Series, which includes texts and monographs covering many topics of current research interest in pure and applied statistics. Each title has an original slant even if the material included is not specificallyoriginal. The authors are leading researchers and the topics covered will be of interest to all professional statisticians, whether they be in industry, government department or research institute. Other books in the series include 23. W.J.Krzanowski: Principles of multivariate analysis: a user'sperspective updated edition 24. J.Durbin and S.J.Koopman: Time series analysis by State Space Models 25. Peter J. Diggle, Patrick Heagerty, Kung-Yee Liang, Scott L. Zeger: Analysis of Longitudinal Data 2/e 26. J.K. Lindsey: Nonlinear Models in Medical Statistics 27. Peter J. Green, Nils L. Hjort andSylvia Richardson: Highly Structured Stochastic Systems 28. Margaret S. Pepe: The Statistical Evaluation of Medical Tests for Classification and Prediction 29. Christopher G. Small and Jinfang Wang: Numerical Methods for Nonlinear Estimating Equations

Preface

xiii

Acknowledgements

xiv

1 Introduction

(10)

1.1 Historical review

(2)

1.2 Current trends

(3)

1.3 Overview of the topics in the book

(2)

1.4 Closed form and algorithmic solutions

(1)

1.5 Notation

(2)

2 Initial transformations

(9)

2.1 Data-scaling of variables

(3)

2.1.1 Canonical analysis as a form of data-scaling

(1)

2.1.2 Prior multivariate analyses

(1)

2.1.3 Distance matrices

(1)

2.2 Configuration-scaling

(1)

2.2.1 Differing numbers of variables

(1)

2.2.2 Isotropic and anisotropic scaling

(1)

2.3 Types of data sets

(4)

3 Two-set Procrustes problems-generalities

(9)

3.1 Introduction

(3)

3.1.1 Least-Squares criteria

(1)

3.1.2 Correlational and inner-product criteria

(1)

3.1.3 Robust criteria

(1)

3.1.4 Quantifications

(1)

3.1.5 Matching of rows and columns

(1)

3.2 Translation

(2)

3.2.1 Simple translation

(1)

3.3 Isotropic scaling

(1)

3.4 The general linear transformation

(1)

3.5 A note on algorithms

(1)

3.6 A K-sets problem with two-sets solutions

(2)

4 Orthogonal Procrustes problems

(19)

4.1 Solution of the orthogonal Procrustes problem

(1)

4.2 Necessary and sufficient conditions for optimality

(1)

4.3 Scaling

(1)

4.4 Example of an orthogonal rotation of two sets, including a scaling factor

(1)

4.5 Different dimensionalities in X1 and X2

(1)

4.6 Constrained orthogonal Procrustes problems

(7)

4.6.1 Rotations and reflections

(2)

4.6.2 The two-dimensional case

(2)

4.6.3 Best Householder reflection

(1)

4.6.4 Best plane rotation

(3)

4.7 Best rank-R fit, principal components analysis and the Eckart-Young theorem

(1)

4.8 Other criteria

(6)

4.8.1 Orthogonal Procrustes by maximising congruence

(1)

4.8.2 Robust orthogonal Procrustes

(5)

5 Projection Procrustes problems

(25)

5.1 Projection Procrustes by inner-product

(1)

5.2 Necessary but not sufficient conditions

(1)

5.3 Two-sided projection Procrustes by inner-product

(6)

5.3.1 Tucker's problem

(1)

5.3.2 Meredith's problem

(3)

5.3.3 Green's problem

(1)

5.4 Projection Procrustes by least squares

(9)

5.4.1 The Kochat and Swayne approach

(2)

5.4.2 The wind model

(5)

5.5 Two-sided projection Procrustes by least-squares

(1)

5.6 Maximising the projected average configuration

(4)

5.6.1 Method 1

(1)

5.6.2 Method 2

(3)

5.7 Rotation into higher dimensions

(1)

5.8 Some geometrical considerations

(4)

6 Oblique Procrustes problems

(11)

6.1 The projection method

(2)

6.2 The parallel axes or vector-sums method

(2)

6.3 The cases T = C-1 and T = (C')-1

(3)

6.4 Summary of results

(1)

7 Other two-sets Procrustes problems

(7)

7.1 Permutations

(1)

7.2 Reduced rank regression

(1)

7.3 Miscellaneous choices of T

(1)

7.4 On simple structure rotations etc

(1)

7.5 Double Procrustes problems

(4)

7.5.1 Double Procrustes for symmetric matrices (orthogonal case)

(1)

7.5.2 Double Procrustes for rectangular matrices (orthogonal case)

(2)

8 Weighting, scaling, and missing values

(12)

8.1 Weighting

(5)

8.1.1 Translation with weighting

(3)

8.1.2 General forms of weighting

(2)

8.2 Missing values

(1)

8.3 Anisotropic scaling

(6)

8.3.1 Pre-scaling R

(1)

8.3.2 Post-scaling S

(1)

8.3.3 Estimation of T with post-scaling S

(2)

8.3.4 Simultaneous estimation of R, S, and T

100

(1)

8.3.5 Scaling with two-sided problems

100

(1)

8.3.6 Row scaling

100

(3)

9 Generalised Procrustes problems

103

(29)

9.1 Results applicable to any transformation

104

(17)

9.1.1 Generalised Procrustes criteria and some basic identities

104

(2)

9.1.2 Algorithms

106

(2)

9.1.3 Missing values without scaling

108

(4)

9.1.4 Generalised Procrustes analysis with isotropic scaling

112

(2)

9.1.5 Contribution of the kth set to the residual sum-of-square

114

(2)

9.1.6 The constraint (9.20) and other possibilities

116

(2)

9.1.7 Missing values with scaling

118

(1)

9.1.8 Exhibiting the group average

119

(2)

9.1.9 Other criteria

121

(1)

9.2 Special transformations

121

(8)

9.2.1 Generalised orthogonal Procrustes analysis

122

(2)

9.2.2 Example of a typical application of GPA in food science

124

(2)

9.2.3 Generalised projection Procrustes analysis

126

(3)

9.3 Generalised pairwise methods

129

(3)

9.3.1 Example of generalised pairwise Procrustes analysis

129

(3)

10 Analysis of variance framework

132

(14)

10.1 Basic formulae

132

(1)

10.2 Geometrical unification of orthogonal and projection Procrustes methods

133

(1)

10.2.1 Analysis of variance of configurations

134

(1)

10.2.2 Partitioning the analysis of variance into components representing projections into independent sub-spaces

137

(1)

10.2.3 Algebraic form of the ANOVA

140

(1)

10.2.4 An example of ANOVA

140

(1)

10.3 Criteria as terms in the ANOVA

141

(1)

10.4 Discussion

142

(4)

11 Incorporating information on variables

146

(10)

11.1 Biplots

147

(1)

11.1.1 Linear biplots

147

(1)

11.1.2 Biplots in multidimensional scaling of transformed dissimilarities

149

(1)

11.1.3 Biplots in multidimensional scaling of transformed variables

150

(1)

11.2 An example: coffee images

151

(5)

12 Accuracy and stability

156

(8)

12.1 Introduction

156

(1)

12.2 Probability distributions of Procrustes loss values

157

(2)

12.3 Resampling methods

159

(1)

12.3.1 Random data method

159

(1)

12.3.2 Random permutation method

161

(1)

12.3.3 Jackknifing

162

(1)

12.4 Dimensionality of results

162

(2)

13 Links with other methods

164

(19)

13.1 Variant treatments of generalised Procrustes analysis

164

(1)

13.1.1 Isotropic scaling of the group average

165

(1)

13.1.2 Anisotropic scaling of the group average

166

(1)

13.1.3 Procrustean individual difference scaling

169

(2)

13.2 Individual differences scaling

171

(1)

13.2.1 Example of INDSCAL

173

(1)

13.3 SMACOF etc.

174

(1)

13.4 Structuration des Tableaux A Trois Indices de la Statisitique

175

(1)

13.5 Links with canonical correlation

176

(5)

13.6 Common components analysis

181

(1)

13.7 Other methods

182

(1)

14 Some application areas, future, and conclusion

183

(12)

14.1 Application areas

183

(1)

14.1.1 Personality psychology/factor analysis

183

(1)

14.1.2 Sensory analysis

184

(1)

14.1.3 Market research

185

(1)

14.1.4 Shape analysis

185

(1)

14.1.5 Biometric identification and warping

189

(1)

14.1.6 Molecular biology

190

(1)

14.1.7 Image analysis

190

(1)

14.1.8 Procrustes methods as a statistical tool

191

(1)

14.2 Note on comparing methods

191

(1)

14.2.1 Comparing criteria

191

(1)

14.2.2 Comparing algorithms

192

(1)

14.2.3 Comparing implementations of algorithms

193

(1)

14.2.4 Comparing models

193

(1)

14.3 A note on software

193

(1)

14.4 Conclusion

194

(1)

Appendix A Configurations

195

(2)

Appendix B Rotations and reflections

197

(6)

B.1 Rotations

197

(2)

B.2 Some special cases

199

(1)

B.3 General orthogonal matrices

200

(3)

Appendix C Orthogonal projections

203

(2)

C.1 Orthonormal operators as rotations

204

(1)

Appendix D Oblique axes

205

(2)

Appendix E A minimisation problem

207

(9)

E.1 Zero values of zi

209

(3)

E.2 Some special cases

212

(2)

E.3 Some remarks on the case k = 0

214

(1)

E.4 Note on an alternative parameterisation

215

(1)

Appendix F Symmetric matrix products

216

(4)

F.1 Special case

218

(2)

References

220

(9)

Index

229

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Procrustes Problems

Summary

Author Biography

Table of Contents

Supplemental Materials