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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. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. 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.
- Sales Rank: #532843 in Books
- Published on: 1990-03
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .43" w x 5.98" l, .70 pounds
- Binding: Paperback
- 180 pages
Review
'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, Short Book Reviews of the International Statistical Institute
'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
' ... 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 Reviews
Most helpful customer reviews
34 of 35 people found the following review helpful.
nonparametric regression and smoothing is basically what splines are all about
By Michael R. Chernick
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.
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