Procrustes problems [electronic resource] / J.C. Gower, G.B. Dijksterhuis.

Gower, J. C.
Oxford : Oxford University Press, 2004.
Oxford statistical science series ; 30.
Oxford statistical science series ; 30
1 online resource (248 p.)
Multivariate analysis.
Electronic books.
Procrustean methods are used to transform one set of data to represent another set of data as closely as possible. This text is a systematic overview of Procrustean methods, presenting a unifying Analysis of Variance framework for different matching methods and the development of statistical tests.
Contents; Preface; Acknowledgements; 1 Introduction; 1.1 Historical review; 1.2 Current trends; 1.3 Overview of the topics in the book; 1.4 Closed form and algorithmic solutions; 1.5 Notation; 2 Initial transformations; 2.1 Data-scaling of variables; 2.1.1 Canonical analysis as a form of data-scaling; 2.1.2 Prior multivariate analyses; 2.1.3 Distance matrices; 2.2 Configuration-scaling; 2.2.1 Differing numbers of variables; 2.2.2 Isotropic and anisotropic scaling; 2.3 Types of data sets; 3 Two-set Procrustes problems-generalities; 3.1 Introduction; 3.1.1 Least-Squares criteria
3.1.2 Correlational and inner-product criteria3.1.3 Robust criteria; 3.1.4 Quantifications; 3.1.5 Matching of rows and columns; 3.2 Translation; 3.2.1 Simple translation; 3.3 Isotropic scaling; 3.4 The general linear transformation; 3.5 A note on algorithms; 3.6 A K-sets problem with two-sets solutions; 4 Orthogonal Procrustes problems; 4.1 Solution of the orthogonal Procrustes problem; 4.2 Necessary and sufficient conditions for optimality; 4.3 Scaling; 4.4 Example of an orthogonal rotation of two sets, including a scaling factor; 4.5 Different dimensionalities in X[sub(1)] and X[sub(2)]
4.6 Constrained orthogonal Procrustes problems4.6.1 Rotations and reflections; 4.6.2 The two-dimensional case; 4.6.3 Best Householder reflection; 4.6.4 Best plane rotation; 4.7 Best rank-R fit, principal components analysis and the Eckart忘oung theorem; 4.8 Other criteria; 4.8.1 Orthogonal Procrustes by maximising congruence; 4.8.2 Robust orthogonal Procrustes; 5 Projection Procrustes problems; 5.1 Projection Procrustes by inner-product; 5.2 Necessary but not sufficient conditions; 5.3 Two-sided projection Procrustes by inner-product; 5.3.1 Tucker's problem; 5.3.2 Meredith's problem
5.3.3 Green's problem5.4 Projection Procrustes by least squares; 5.4.1 The Kochat and Swayne approach; 5.4.2 The wind model; 5.5 Two-sided projection Procrustes by least-squares; 5.6 Maximising the projected average configuration; 5.6.1 Method 1; 5.6.2 Method 2; 5.7 Rotation into higher dimensions; 5.8 Some geometrical considerations; 6 Oblique Procrustes problems; 6.1 The projection method; 6.2 The parallel axes or vector-sums method; 6.3 The cases T = C[sup(-1)] and T = (C')[sup(-1)]; 6.4 Summary of results; 7 Other two-sets Procrustes problems; 7.1 Permutations; 7.2 Reduced rank regression
7.3 Miscellaneous choices of T7.4 On simple structure rotations etc; 7.5 Double Procrustes problems; 7.5.1 Double Procrustes for symmetric matrices (orthogonal case); 7.5.2 Double Procrustes for rectangular matrices (orthogonal case); 8 Weighting, scaling, and missing values; 8.1 Weighting; 8.1.1 Translation with weighting; 8.1.2 General forms of weighting; 8.2 Missing values; 8.3 Anisotropic scaling; 8.3.1 Pre-scaling R; 8.3.2 Post-scaling S; 8.3.3 Estimation of T with post-scaling S; 8.3.4 Simultaneous estimation of R, S, and T; 8.3.5 Scaling with two-sided problems; 8.3.6 Row scaling
9 Generalised Procrustes problems
Description based upon print version of record.
Includes bibliographical references (p. [220]-227) and index.
Description based on print version record.
Dijksterhuis, Garmt B.
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