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Spline Models for Observational Data

Grace Wahba

CBMS-NSF Regional Conference Series in Applied Mathematics 59

"This is a thorough account of non-parametric regression using splines, eschewing other approaches, and approaching splines themselves via the technology of reproducing kernel Hilbert spaces. The result is an impressively unified, consistent, treatment of a wide variety of problems, some really quite hard...This is an impressive record of research, offering stimulation for further investigation." -- P.J. Green, University of Bristol, Short Book Reviews, Volume 10, Number 3, December 1990.

"The book provides a rather complete unified treatment of smoothing splines, starting with the classical polynomial smoothing spline, and including the periodic smoothing spline on a circle, both scalar and vector-valued splines on the sphere, and thin plate splines in the plane and in higher dimensional Euclidean spaces. In addition, it treats two special kinds of smoothing splines called partial splines and additive splines. The splines discussed here have numerous practical applications in data fitting of economical, medical, meteorological, and radiation data. She provides applications to the solution of Fredholm integral equations of the first kind, fluid flow problems in porous media, and certain inverse problems." -- Larry L. Schumaker, Vanderbilt University, SIAM Review, June 1991.

"...The reviewer considers the monograph a valuable contribution and recommends it strongly to everyone with some interest in this important area of statistics." -- Girdhar G. Agarwal, Mathematical Review, Issue 91G

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.


Foreword; Chapter 1: Background; Chapter 2: More Splines; Chapter 3: Equivalence and Perpendicularity, or, What's So Special About Splines?; Chapter 4: Estimating the Smoothing Parameter; Chapter 5: "Confidence Intervals"; Chapter 6: Partial Spline Models; Chapter 7: Finite-Dimensional Approximating Subspaces; Chapter 8: Fredholm Integral Equations of the First Kind; Chapter 9: Further Nonlinear Generalizations; Chapter 10: Additive and Interaction Splines; Chapter 11: Numerical Methods; Chapter 12: Special Topics; Bibliography; Author Index.

Adopted for classroom use at Rice University.

1990 / xii + 169 pages / Softcover / ISBN-13: 978-0-898712-44-5 / ISBN-10: 0-89871-244-0 /
List Price $60.50 / SIAM/CBMS Member Price $42.35 / Order Code CB59
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Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Auckland, NZ.)