The Computation of Generalized Cross-Validation Functions Through Householder Tridiagonalization with Applications to the Fitting of Interaction Spline Models

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Abstract
An efficient algorithm for computing the GCV (generalized cross-validation) function for the general cross-validated regularization/smoothing problem is provided. This algorithm is based on the Householder tridiagonalization, similar to Elden’s [BIT, 24 (1984), pp. 467–472] bidiagonalization and is appropriate for problems where no natural structure is available, and the regularization /smoothing problem is solved (exactly) in a reproducing kernel Hilbert space. It is particularly appropriate for certain multivariate smoothing problems with irregularly spaced data, and certain remote sensing problems, such as those that occur in meteorology, where the sensors are arranged irregularly.The algorithm is applied to the fitting of interaction spline models with irregularly spaced data and two smoothing parameters, and favorable timing results are presented. The algorithm may be extended to the computation of certain GML (generalized maximum likelihood) functions. Application of the GML algorithm

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