Splines in Nonparametric Regression

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. It then ends with some remarks on computing. Publicly available software is mentioned along the way.