Cross-validation techniques for smoothing spline functions in one or two dimensions

Abstract

This paper aims to present some results on the asymptotic behaviour of a matrix associated with certain types of spline functions and shows how these results can be used to obtain a fast algorithm for choosing the smoothing parameter in the smoothing of noisy data by splines.First, we give a general theorem on the behaviour of the eigenvalues associated with a spline function. The spline functions considered here are those defined by a variational formulation (for general theorems on these splines see [1], [11]).Second, we use the preceding results to obtain an O(n) — algorithm to calculate the optimal smoothing polynomial spline using the generalized Cross-Validation Technique, in the case of equally spaced data. We obtain an O(n2) algorithm to perform this calculation in the case of non equally spaced data. Some very good numerical results are presented and a table of run times is given. We then introduce a method which calculates the solution (smoothed data) by a piecewise smoothing and correction technique. This method allows us to treat a very important amount of data.We show too, how calculations can be made with two dimensional data (with arbitrary data points) and generalize the piecewise calculations to this case. We also present a set of numerical results and a table of run times.KeywordsNull SpaceSpline FunctionThin Plate SplinePolynomial SplinePreceding SubsectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.