An iterative procedure for calculating minimum generalised cross validation smoothing splines

P. A. Hancock, M. F. Hutchinson


This study analyses a simple iterative procedure for estimating minimum generalised cross-validation (GCV) univariate smoothing splines. The results provide guidelines for the development of a similar methodology to estimate minimum GCV bivariate thin plate smoothing splines. The methodology is based on the techniques described in Hutchinson [ ANZIAM J , 42(E):C774--C796, 2000], which uses nested grid SOR iterative methods in order to solve finite element thin plate smoothing spline systems efficiently for large data sets. The method also uses the stochastic approximation to the GCV developed by Hutchinson [ Commun. Stats --- Sim. and Comp. , 18:1059--1076, 1989]. A double iteration is used to produce increasingly accurate estimates of the minimum GCV smoothing parameter and the smoothing spline. First and second derivatives of the GCV with respect to the smoothing parameter are used to update the smoothing parameter. Convergence of the SOR iteration is improved significantly by correcting the solution estimate after each smoothing parameter update using the estimate of the derivative of the solution with respect to the smoothing parameter.

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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.