## Procrustes problems.(English)Zbl 1057.62044

Oxford Statistical Science Series 30. Oxford: Oxford University Press (ISBN 0-19-851058-6/hbk). xiv, 233 p. (2004).
This is the first monograph on Procrustes methods. Procrustes analysis originated from psychometrics, especially from factor analysis. Nowadays, it is applied in molecular biology, sensory analysis, image analysis, etc. Given matrices $$X_1$$ and $$X_2$$, the Procrustes problem (PP) seeks a matrix $$T$$ that minimises the Frobenius norm $$\| X_1T-X_2\|_F$$. Three cases are considered, where $$T$$ are orthogonal, projection or oblique axes transformations. Next, the two sets PP are considered, $$\| X_1T_1- X_2T_2\|_F \to\min$$, and $$\| SX_1T-X_2\|_F \to\min$$. Ch. 8 deals with weighting of rows or columns of data matrices. In particular, missing values may be specified by giving zero weights. Ch. 9 is concerned with generalizations where the two sets of configurations $$X_1$$ and $$X_2$$ are replaced by $$k$$ sets, $$X_1,\dots, X_k$$, each with its own transformation matrix $$T_1,\dots,T_k$$. The group average configuration $$G=k^{-1} \sum^k_{j=1} X_jT_j$$ plays a central role.
Ch. 10 presents analysis of variance in the multi-set problem. Terms in the analysis of variance throw more light on the possible choice of criteria suitable for fitting Procrustes models by least squares. Biplot methodology, which is dicussed in Ch. 11, makes it possible to incorporate information on variables. Ch. 12 contains a brief review of probability models for PP. Ch. 13 gives links of the multi-set PP with other three-mode scaling methods. Some application areas are indicated in Ch. 14.
Contents: Ch. 1. Introduction; Ch. 2. Initial transformations; Ch. 3. Two-set PP – generalities; Ch. 4. Orthogonal PP; Ch. 5. Projection PP; Ch. 6. Oblique PP; Ch. 7. Other two-sets PP; Ch. 8. Weighting, scaling, and missing values; Ch. 9. Generalised PP; Ch. 10. Analysis of variance framework; Ch. 11. Incorporating information on variables; Ch. 12. Accuracy and stability; Ch. 13. Links with other methods; Ch. 14. Some application areas, future, and conclusions.
Note of the reviewer. PP corresponds to a multivariate multiple regression $$X^0_1T= X_2^0$$, $$X_2=X_2^0+\widetilde X_2$$, where $$X_1^0$$ and $$X_2^0$$ are non-stochastic matrices, $$\widetilde X_2$$ is the random error matrix, $$X_1^0$$ and $$X_2$$ are observed, and $$P$$ is to be estimated. If instead of $$X^0_1$$ the surrogate $$X_1= X_1^0+\widetilde X_1$$ is observed, then we have the Total Least Squares (TLS) problem, see S. van Huffel and J. Vandewalle [The total least squares problem: computational aspects and analysis. (1991; Zbl 0789.62054)]. Under additional Hankel/Toeplitz relations in the data matrix $$[X_1,X_2]$$, we have the Structured TLS problem, see I. Markovsky, S. Van Huffel and A. Kukush, Numer. Linear Algebra Appl. 11, 591–608 (2004).

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62H12 Estimation in multivariate analysis 65F20 Numerical solutions to overdetermined systems, pseudoinverses

Zbl 0789.62054
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