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Splines in Nonparametric Regression

Statistical and Numerical Computing

  1. Hao Helen Zhang

Published Online: 15 JAN 2013

DOI: 10.1002/9780470057339.vas052.pub2

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Helen Zhang, H. 2013. Splines in Nonparametric Regression . Encyclopedia of Environmetrics. 5.

Author Information

  1. The University of Arizona, AZ, USA

Publication History

  1. Published Online: 15 JAN 2013

Abstract

This article is interested in splines as tools for visualizing and analyzing noisy observational data, and so restricts itself to smoothing splines and regression splines. The article first describes the univariate polynomial smoothing spline, which may be thought of as the forerunner of spline functions used in data analysis. It then describes cross-validation and generalized cross-validation (GCV) for choosing the smoothing parameter. After briefly describing regression splines, this entry then describes a number of generalizations of the univariate smoothing spline to various domains, which are obtained via the solution of a variational problem. These include the thin plate spline, the histospline, splines on the sphere, vector splines on the sphere, hybrid splines, partial splines, and smoothing spline analysis of variance (ANOVA) models on complex domains. Computational issues are also discussed. The last two sections focus on the topics of variable selection and model selection in nonparametric regression. Various penalized regression with penalty for function shrinkage and selection are described. Publicly available software is mentioned along the way.

Keywords:

  • smoothing;
  • generalized cross-validation;
  • shrinkage;
  • variable selection;
  • model selection