Procrustes Problems

by Gower, John C.; Dijksterhuis, Garmt B

• ISBN13: 9780198510581
• ISBN10: 0198510586
• Format: Hardcover
• Copyright: 2004-04-08
• Publisher: Oxford University Press

Summary

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

Author Biography

John C. Gower is on the Editorial board of Journal of Classification and of The Mathematical Scientist. He is a past President of the International Federation of Classification Societies, a past President of the British Region of the International Biometric society, and a past Honorary Secretary of the Royal Statistical Society.
Garmt B. Dijksterhuis is on the Editorial board of Food Quality and Preference and is the Chairman of Sensometric Society.

Table of Contents

Preface	xiii
Acknowledgements	xiv
1 Introduction	1	(10)
1.1 Historical review	2	(2)
1.2 Current trends	4	(3)
1.3 Overview of the topics in the book	7	(2)
1.4 Closed form and algorithmic solutions	9	(1)
1.5 Notation	9	(2)
2 Initial transformations	11	(9)
2.1 Data-scaling of variables	12	(3)
2.1.1 Canonical analysis as a form of data-scaling	13	(1)
2.1.2 Prior multivariate analyses	14	(1)
2.1.3 Distance matrices	14	(1)
2.2 Configuration-scaling	15	(1)
2.2.1 Differing numbers of variables	15	(1)
2.2.2 Isotropic and anisotropic scaling	16	(1)
2.3 Types of data sets	16	(4)
3 Two-set Procrustes problems-generalities	20	(9)
3.1 Introduction	20	(3)
3.1.1 Least-Squares criteria	20	(1)
3.1.2 Correlational and inner-product criteria	21	(1)
3.1.3 Robust criteria	22	(1)
3.1.4 Quantifications	22	(1)
3.1.5 Matching of rows and columns	23	(1)
3.2 Translation	23	(2)
3.2.1 Simple translation	24	(1)
3.3 Isotropic scaling	25	(1)
3.4 The general linear transformation	26	(1)
3.5 A note on algorithms	26	(1)
3.6 A K-sets problem with two-sets solutions	27	(2)
4 Orthogonal Procrustes problems	29	(19)
4.1 Solution of the orthogonal Procrustes problem	30	(1)
4.2 Necessary and sufficient conditions for optimality	31	(1)
4.3 Scaling	32	(1)
4.4 Example of an orthogonal rotation of two sets, including a scaling factor	32	(1)
4.5 Different dimensionalities in X1 and X2	33	(1)
4.6 Constrained orthogonal Procrustes problems	34	(7)
4.6.1 Rotations and reflections	34	(2)
4.6.2 The two-dimensional case	36	(2)
4.6.3 Best Householder reflection	38	(1)
4.6.4 Best plane rotation	38	(3)
4.7 Best rank-R fit, principal components analysis and the Eckart-Young theorem	41	(1)
4.8 Other criteria	42	(6)
4.8.1 Orthogonal Procrustes by maximising congruence	42	(1)
4.8.2 Robust orthogonal Procrustes	43	(5)
5 Projection Procrustes problems	48	(25)
5.1 Projection Procrustes by inner-product	49	(1)
5.2 Necessary but not sufficient conditions	49	(1)
5.3 Two-sided projection Procrustes by inner-product	50	(6)
5.3.1 Tucker's problem	51	(1)
5.3.2 Meredith's problem	52	(3)
5.3.3 Green's problem	55	(1)
5.4 Projection Procrustes by least squares	56	(9)
5.4.1 The Kochat and Swayne approach	58	(2)
5.4.2 The wind model	60	(5)
5.5 Two-sided projection Procrustes by least-squares	65	(1)
5.6 Maximising the projected average configuration	65	(4)
5.6.1 Method 1	66	(1)
5.6.2 Method 2	66	(3)
5.7 Rotation into higher dimensions	69	(1)
5.8 Some geometrical considerations	69	(4)
6 Oblique Procrustes problems	73	(11)
6.1 The projection method	76	(2)
6.2 The parallel axes or vector-sums method	78	(2)
6.3 The cases T = C-1 and T = (C')-1	80	(3)
6.4 Summary of results	83	(1)
7 Other two-sets Procrustes problems	84	(7)
7.1 Permutations	84	(1)
7.2 Reduced rank regression	85	(1)
7.3 Miscellaneous choices of T	86	(1)
7.4 On simple structure rotations etc	87	(1)
7.5 Double Procrustes problems	87	(4)
7.5.1 Double Procrustes for symmetric matrices (orthogonal case)	88	(1)
7.5.2 Double Procrustes for rectangular matrices (orthogonal case)	89	(2)
8 Weighting, scaling, and missing values	91	(12)
8.1 Weighting	91	(5)
8.1.1 Translation with weighting	91	(3)
8.1.2 General forms of weighting	94	(2)
8.2 Missing values	96	(1)
8.3 Anisotropic scaling	97	(6)
8.3.1 Pre-scaling R	97	(1)
8.3.2 Post-scaling S	98	(1)
8.3.3 Estimation of T with post-scaling S	98	(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