Procrustes problems [electronic resource] / J.C. Gower, G.B. Dijksterhuis.
- 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. -227) and index.
Description based on print version record.
- Dijksterhuis, Garmt B.
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