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On Generalized Cross Validation for Tensor Smoothing Splines

The natural tensor-product smoothing spline is one of the methods of choice for fitting noisy data given on a grid. A generalized cross-validation procedure for automatic selection of the smoothing parameter in the method is introduced. It is shown that as in the well-known univariate and thin plate spline cases, the method selects the parameter in an asymptotically optimal way. Computational aspects of the method are also discussed, and a numerical example is presented. The method developed here can also be extended to complete and periodic tensor-product smoothing splines.