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2012 The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy
Nancy Heckman
Statist. Surv. 6: 113-141 (2012). DOI: 10.1214/12-SS101

Abstract

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_{j}(Y_{j}-\mu(t_{j}))^{2}+\lambda \int_{a}^{b}[\mu''(t)]^{2}\,dt$, where the data are $t_{j},Y_{j}$, $j=1,\ldots,n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function $\mu$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, $\int_{a}^{b}[\mu''(t)]^{2}\,dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to the construction and study of the Reproducing Kernel Hilbert Space corresponding to a penalty based on a linear differential operator. In this case, one can often calculate the minimizer explicitly, using Green’s functions.

Citation

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Nancy Heckman. "The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy." Statist. Surv. 6 113 - 141, 2012. https://doi.org/10.1214/12-SS101

Information

Published: 2012
First available in Project Euclid: 16 October 2012

zbMATH: 1302.62165
MathSciNet: MR3011073
Digital Object Identifier: 10.1214/12-SS101

Subjects:
Primary: 46E22, 62G99
Secondary: 62G08

Rights: Copyright © 2012 The author, under a Creative Commons Attribution License

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