Spline Models for Observational Data
Seiten
1990
Society for Industrial & Applied Mathematics,U.S. (Verlag)
9780898712445 (ISBN)
Society for Industrial & Applied Mathematics,U.S. (Verlag)
9780898712445 (ISBN)
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Peovides an introduction into the more theoretical aspects of the use of spline models. This book develops a theory and practice for the estimation of functions from noisy data on functionals.
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.
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.
| Erscheint lt. Verlag | 30.9.1990 |
|---|---|
| Reihe/Serie | CBMS-NSF Regional Conference Series in Applied Mathematics |
| Verlagsort | New York |
| Sprache | englisch |
| Maße | 171 x 247 mm |
| Gewicht | 340 g |
| Themenwelt | Mathematik / Informatik ► Mathematik |
| ISBN-13 | 9780898712445 / 9780898712445 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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