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A robust approach to nonlinear multivariate analysis. (English) Zbl 0868.62052

Leiden: D.S.W.O. Press. 191 p. (1994).
The title of the book is attractive. The bock includes seven chapters: Robustness and related concepts; Robust loss functions and iterative majorization; Orthogonal Procrustes analysis; Nonlinear multiple regression analysis; Canonical distribution analysis: Lower rank approximation of matrices; and Epilogue.
The book includes typical topics covered in the study of the effects of contaminated data. The analysis is based on some alternative loss criteria for least squares, especially for Huber’s loss function and Tukey’s biweight loss function. After the introduction and discussion of some basic concepts, the book focuses on the orthogonal Procrustes analysis, nonlinear multiple regression analysis, canonical discriminant analysis and principal components analysis. As the most basic form in the Procrustes problem, an orthogonal rotation to match one configuration with another, the author paid most attention to handle the outliers arising in the Procrustes problem. An algorithm for robust Procrustes analysis was proposed. This algorithm, including vectorwise weighting approach as well as elementwise weighting approach, is useful in practice. Although the approach proposed in this book seems more complicated for \(p\) \((>2)\) configurations, I do not know whether there are other better methods.
On the robustness in linear regression, the generalized \(M\)-estimators have been developed. Some estimators were extended to apply in nonlinear regression. The author has discussed the applications of Schweppes and Mallows type estimators in the nonlinear situation in detail. In terms of illustrative examples and simulations, it is shown that the Huber and biweight \(M\)-estimators are also useful in nonlinear multiple regression. Some techniques to overcome the outliers in nonlinear multiple regression analysis are also suggested in Chapter 4.
In order to overcome the influence of outliers in canonical discriminant analysis, the Huber and biweight loss functions are used and the corresponding robust procedures are discussed. Especially, the algorithms in Chapter 5 are useful in practice. “Although a few percent does not seem much, one could easily think of real life situations in which a few percent improvement of prediction could have an enormous practical impact”, as the author wrote.
The principal components analysis is an important technique to represent high-dimensional data in a low-dimensional space. Data sets with outliers analyzed by principal components analysis are studied in Chapter 6. By means of the robust framework instead of the least squares framework, some approaches for the robust lower rank approximation, such as via the covariance matrix, the Huber or biweight loss function and optimal scaling have been analyzed. For real life data sets and simulation data sets various approaches have been studied and compared on linear and nonlinear situations.
The author gave a minute description of the related concepts and techniques in data analysis. With a plenty of illustrations, the theory and methods are easy to comprehend. The plenty of text covers the main themes on outliers arising in multivatiate analysis. The iterative majorization useful for constructing convergent algorithms has been extensively used in this book. As well, related approaches in this field have been compared and may be generalized to further study.
In summary, I would say that this book is well-organized. It would be even better if the author gave more practical examples. The book is appropriate as a reference book for graduate students in statistics.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62F35 Robustness and adaptive procedures (parametric inference)
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62J02 General nonlinear regression
62H99 Multivariate analysis
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