Optimal Smoothing of Noisy Data Using Spline Functions

Abstract

We consider the problem of approximating a function f supposed to be “smooth”, given its values known with error at n different points of a real interval $[a,b]$. To approximate f we use the natural smoothing spline of order q and parameter $\tau $. For choosing $\tau $, the method of generalized cross validation, proposed by Wahba and others, has very interesting theoretical properties, but requires expensive calculations. The asymptotic behavior of the eigenvalues associated with the spline functions provides a practical method for calculating the GCV function which reduces the computation time by a factor of n.