Smoothing Splines

Methods and Applications

By Yuedong Wang

© 2011 – Chapman and Hall/CRC

394 pages | 94 B/W Illus.

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Hardback: 9781420077551
pub: 2011-06-21
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About the Book

A general class of powerful and flexible modeling techniques, spline smoothing has attracted a great deal of research attention in recent years and has been widely used in many application areas, from medicine to economics. Smoothing Splines: Methods and Applications covers basic smoothing spline models, including polynomial, periodic, spherical, thin-plate, L-, and partial splines, as well as more advanced models, such as smoothing spline ANOVA, extended and generalized smoothing spline ANOVA, vector spline, nonparametric nonlinear regression, semiparametric regression, and semiparametric mixed-effects models. It also presents methods for model selection and inference.

The book provides unified frameworks for estimation, inference, and software implementation by using the general forms of nonparametric/semiparametric, linear/nonlinear, and fixed/mixed smoothing spline models. The theory of reproducing kernel Hilbert space (RKHS) is used to present various smoothing spline models in a unified fashion. Although this approach can be technical and difficult, the author makes the advanced smoothing spline methodology based on RKHS accessible to practitioners and students. He offers a gentle introduction to RKHS, keeps theory at a minimum level, and explains how RKHS can be used to construct spline models.

Smoothing Splines offers a balanced mix of methodology, computation, implementation, software, and applications. It uses R to perform all data analyses and includes a host of real data examples from astronomy, economics, medicine, and meteorology. The codes for all examples, along with related developments, can be found on the book’s web page.


A distinguished strength of this book is the wide variety of real data sets used to illustrate models and methods. … extremely helpful for practitioners … For each method, the book provides all the necessary computational details, including explicit formulae and detailed algorithms. … It is an ideal textbook for a high-level graduate student course and an ideal reference for those who deal with complicated nonparametric or semiparametric regression models. … I think this is a great book on smoothing splines that one should treasure like Wahba and Gu.

—Pang Du, Biometrics, December 2012

… a readable text that focuses on methodology, computation, implementation, software, and application. The book is lavishly illustrated with real examples and incorporates many figures which clearly demonstrate the differences between the various smoothing spline models far more effectively than mere words could ever do. A library implemented in the R language is available to apply the methods described, and the analyses undertaken, in the book. For anyone wishing to explore the utility of smoothing spline models and the ease with which they can be fitted and explored, I recommend this text as your first reference before delving into the technical details of the underlying RKHS.

International Statistical Review, 80, 2012

This excellent book aims at making the advanced smoothing spline methodology based on reproducing kernel Hilbert spaces (RKHS) more accessible to practitioners and students. It provides software and examples to enable spline smoothing methods to be routinely used in practice … The exposition is very clear; the author takes great care to motivate the different tools and to explain their use. When there are different approaches for the same problem, their pros and cons are carefully considered. Throughout the book, the systematic use of RKHS helps the reader to understand the main issues. The book can be used as reference book and also serve as a text for an advanced course.

—Ricardo Maronna, Statistical Papers, September 2012

Table of Contents


Parametric and Nonparametric Regression

Polynomial Splines

Scope of This Book

The assist Package

Smoothing Spline Regression

Reproducing Kernel Hilbert Space

Model Space for Polynomial Splines

General Smoothing Spline Regression Models

Penalized Least Squares Estimation

The ssr Function

Another Construction for Polynomial Splines

Periodic Splines

Thin-Plate Splines

Spherical Splines

Partial Splines


Smoothing Parameter Selection and Inference

Impact of the Smoothing Parameter


Unbiased Risk

Cross-Validation and Generalized Cross-Validation

Bayes and Linear Mixed-Effects Models

Generalized Maximum Likelihood

Comparison and Implementation

Confidence Intervals

Hypothesis Tests

Smoothing Spline ANOVA

Multiple Regression

Tensor Product Reproducing Kernel Hilbert Spaces

One-Way SS ANOVA Decomposition

Two-Way SS ANOVA Decomposition

General SS ANOVA Decomposition

SS ANOVA Models and Estimation

Selection of Smoothing Parameters

Confidence Intervals


Spline Smoothing with Heteroscedastic and/or Correlated Errors

Problems with Heteroscedasticity and Correlation

Extended SS ANOVA Models

Variance and Correlation Structures


Generalized Smoothing Spline ANOVA

Generalized SS ANOVA Models

Estimation and Inference

Wisconsin Epidemiological Study of Diabetic Retinopathy

Smoothing Spline Estimation of Variance Functions

Smoothing Spline Spectral Analysis

Smoothing Spline Nonlinear Regression


Nonparametric Nonlinear Regression Models

Estimation with a Single Function

Estimation with Multiple Functions

The nnr Function


Semiparametric Regression


Semiparametric Linear Regression Models

Semiparametric Nonlinear Regression Models


Semiparametric Mixed-Effects Models

Linear Mixed-Effects Models

Semiparametric Linear Mixed-Effects Models

Semiparametric Nonlinear Mixed-Effects Models


Appendix A: Data Sets

Appendix B: Codes for Fitting Strictly Increasing Functions

Appendix C: Codes for Term Structure of Interest Rates


Author Index

Subject Index

About the Author

Yuedong Wang is a professor and the chair of the Department of Statistics and Applied Probability at the University of California–Santa Barbara. Dr. Wang is an elected fellow of the ASA and ISI, a fellow of the RSS, and a member of IMS, IBS, and ICSA. His research covers the development of statistical methodology and its applications.

About the Series

Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General