Spline models for observational data

Grace Wahba9780898712445, 0898712440

This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. The estimate is a polynomial smoothing spline. By placing this smoothing problem in the setting of reproducing kernel Hilbert spaces, a theory is developed which includes univariate smoothing splines, thin plate splines in d dimensions, splines on the sphere, additive splines, and interaction splines in a single framework. A straightforward generalization allows the theory to encompass the very important area of (Tikhonov) regularization methods for ill-posed inverse problems.
Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a wide variety of problems which fall within this framework. Methods for including side conditions and other prior information in solving ill-posed inverse problems are included. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.

---

Table of contents :
Spline Models for Observational Data……Page 1
Contents……Page 10
Foreword……Page 12
CHAPTER 1 Background……Page 18
CHAPTER 2 More Splines……Page 38
CHAPTER 3 Equivalence and Perpendicularity, or,What’s So Special About Splines?……Page 58
CHAPTER 4 Estimating the Smoothing Parameter……Page 62
CHAPTER 5 “Confidence Intervals”……Page 84
CHAPTER 6 Partial Spline Models……Page 90
CHAPTER 7 Finite-Dimensional Approximating Subspaces……Page 112
CHAPTER 8 Fredholm Integral Equations of the First Kind……Page 118
CHAPTER 9 Further Nonlinear Generalizations……Page 126
CHAPTER 10 Additive and Interaction Splines……Page 144
CHAPTER 1 1 Numerical Methods……Page 152
CHAPTER 12 Special Topics……Page 162
Bibliography……Page 170
Author Index……Page 184