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
Keywords
SmoothingSmoothing splineRegularization (linguistics)Cross-validationComputationMathematicsAlgorithmSpline (mechanical)Kernel (algebra)Reproducing kernel Hilbert spaceMathematical optimizationBasis functionHilbert spaceApplied mathematicsComputer scienceSpline interpolationArtificial intelligenceCombinatoricsMathematical analysisStatistics
Affiliated Institutions
Related Publications
Minimizing GCV/GML Scores with Multiple Smoothing Parameters via the Newton Method
The (modified) Newton method is adapted to optimize generalized cross validation (GCV) and generalized maximum likelihood (GML) scores with multiple smoothing parameters. The ma...
The Performance of Cross-Validation and Maximum Likelihood Estimators of Spline Smoothing Parameters
Abstract An important aspect of nonparametric regression by spline smoothing is the estimation of the smoothing parameter. In this article we report on an extensive simulation s...
Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion
Summary Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoo...
Cross-Validating Non-Gaussian Data
Abstract This article describes an appropriate way of implementing the generalized cross-validation method and some other least-squares-based smoothing parameter selection metho...
Component selection and smoothing in multivariate nonparametric regression
We propose a new method for model selection and model fitting in multivariate nonparametric regression models, in the framework of smoothing spline ANOVA. The “COSSO ” is a meth...
Publication Info
- Year
- 1989
- Type
- article
- Volume
- 10
- Issue
- 4
- Pages
- 457-480
- Citations
- 80
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
80
OpenAlex
1
Influential
56
CrossRef
Cite This
Chong Gu,
Douglas M. Bates,
Zehua Chen
et al.
(1989).
The Computation of Generalized Cross-Validation Functions Through Householder Tridiagonalization with Applications to the Fitting of Interaction Spline Models.
SIAM Journal on Matrix Analysis and Applications
, 10
(4)
, 457-480.
https://doi.org/10.1137/0610033
Identifiers
- DOI
- 10.1137/0610033