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Smoothing splines. Methods and applications. (English) Zbl 1223.65011

Monographs on Statistics and Applied Probability 121. Boca Raton, FL: CRC Press (ISBN 978-1-4200-7755-1/hbk; 978-1-032-47762-6/pbk; 978-1-4200-7756-8/ebook). xxiv, 370 p. (2011).
Smoothing splines provide one of the most successful and most often used methods for the approximation of given data. There is a variety of different approaches which differ, e.g., by the (semi-)norm which is chosen to control the smoothness of the approximation, by the approximation space or by the way parameters are fixed, for instance by generalised cross-validation. In this book, a large collection of different smoothing spline approaches is discussed, which includes a great many examples with practical data. All of the basic theory is explained which includes, of course, reproducing kernel Hilbert spaces, penalised least squares, radial basis functions (in the form of thin-plate splines), \(L\)-splines – with the various forms in which they appear in applications (for instance exponential splines or trigonometric splines). All this is to be found in the first two chapters.
Further material is provided on the mentioned choice of smoothing parameters, most importantly cross-validation and generalised cross-validation (Chapter 3). The following chapters concern certain particular aspects of smoothing splines, namely smoothing spline analysis of variance with many examples (Chapters 4–6) and regression (Chapters 7 and 8), again with many practical examples. Examples and data sets are mentioned in detail, incidentally, in Appendix A and there are C and R codes in the following appendices. The final chapter is about linear and nonlinear mixed effects models using smoothing splines. Codes for implementing the discussed methods and numerical examples can be found not only in the appendix but throughout the book which makes it a useful contribution for everybody who wishes to employ smoothing splines for approximation.

MSC:

65D07 Numerical computation using splines
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
41A15 Spline approximation
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
62J05 Linear regression; mixed models
65D10 Numerical smoothing, curve fitting
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