Product Description
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.

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nonparametric regression and smoothing is basically what splines are all about

As a student of Manny Parzen at Stanford Grace Wahba worked in the area of reproducing kernel Hilbert Space and cubic spline smoothing. Basically splines are special flexible functions that can be used to fit regression functions to date without assuming a linear or fixed degree polynomial. It is the pasting together of local polynomial functions (e.g. cubic functions) where the polynomial changes definitions at a set of points called the knots of the spline. To maintain a smoothness to the function the constraints are placed on the derivatives of the splines at the knors. This is intended to give them continuity and smoothness at the points of connection.

In this monograph Grace Wahba describes how to construct and fit such splines to data. In so doing smoothness, goodness of fit and ability ot predict are the important attributes. Appropriate loss functions with smoothness constraints are used in the fit. The number and location of the knots can be fixed or it can ve determined based on the sample data. It is important to note that to determine whether the spline is a goof predictor techniques such as cross-validation are required.
March 27, 2008

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