Practical Smoothing: The Joys of P-splines

Description

This is a practical guide to P-splines, a simple, flexible and powerful tool for smoothing. P-splines combine regression on B-splines with simple, discrete, roughness penalties. They were introduced by the authors in 1996 and have been used in many diverse applications. The regression basis makes it straightforward to handle non-normal data, like in generalized linear models. The authors demonstrate optimal smoothing, using mixed model technology and Bayesian estimation, in addition to classical tools like cross-validation and AIC, covering theory and applications with code in R. Going far beyond simple smoothing, they also show how to use P-splines for regression on signals, varying-coefficient models, quantile and expectile smoothing, and composite links for grouped data. Penalties are the crucial elements of P-splines; with proper modifications they can handle periodic and circular data as well as shape constraints. Combining penalties with tensor products of B-splines extends these attractive properties to multiple dimensions. An appendix offers a systematic comparison to other smoothers. Copyright Dedication Contents Preface 1 Introduction 2 Bases, Penalties, and Likelihoods 2.1 Linear and Polynomial Regression 2.2 B-splines 2.3 Penalized Least Squares 2.4 Interpolation and Extrapolation 2.5 Derivatives 2.6 The Effective Dimension 2.7 Standard Errors 2.8 Heavy Smoothing and Polynomial Limits 2.9 P-splines as a Parametric Model 2.10 Whittaker: P-splines without B-splines 2.11 Equivalent Kernels 2.12 Smoothing of a Non-normal Response 2.12.1 Poisson Smoothing 2.12.2 Binomial Smoothing 2.12.3 GLM Effective Dimension and Standard Errors 2.13 Notes and Details 3 Optimal Smoothing in Action 3.1 Cross-Validation 3.2 Akaike’s Information Criterion 3.3 Density Estimation 3.4 Mixed Models 3.5 Bayesian P-splines 3.6 Dangers of Automatic Smoothing 3.7 L- and V-curves 3.8 Transformation of the Independent Variable 3.9 Notes and Details 4 Multidimensional Smoothing 4.1 Generalized Additive Models 4.2 Varying Coefficient Models 4.3 Tensor Product Models 4.4 Tensor Product Bases 4.5 Two-Dimensional Penalties 4.6 Interpolation and Extrapolation 4.7 Smoothing on Large Grids 4.8 Generalized Two-Dimensional Smoothing 4.9 Optimal Two-Dimensional Smoothing 4.10 Issues with Isotropic Smoothing 4.11 Higher Dimensions 4.12 Nested Bases and PS-ANOVA 4.13 Notes and Details 5 Smoothing of Scale and Shape 5.1 Quantile Smoothing 5.2 Expectile Smoothing 5.3 Models for Shape and Scale Parameters 5.4 Baseline Estimation 5.5 Notes and Details 6 Complex Counts and Composite Links 6.1 Histograms with Wide Bins 6.2 Histograms and Scale Transformation 6.3 Individual Censoring 6.4 Latent Mixtures 6.5 Notes and Details 7 Signal Regression 7.1 A Chemical Calibration Problem 7.2 Extensions to the Generalized Linear Model 7.3 Multidimensional Signal Regression 7.4 Further Extensions 7.5 Notes and Details 8 Special Subjects 8.1 The Proper B-spline Basis 8.2 Harmonic Smoothing 8.3 Circular Smoothing 8.4 Signal Separation with Penalties 8.5 Double Penalties 8.6 Piecewise Constant Smoothing 8.7 Shape Constraints 8.8 Variable and Adaptive Penalties 8.9 Survival Analysis and Mortality Modeling 8.10 Notes and Details Appendix A P-splines for the Impatient Appendix B P-splines and Competitors Appendix C Computational Details Appendix D Array Algorithms Appendix E Mixed Model Equations Appendix F Standard Errors in Detail Appendix G The Website References Index