Explicit evaluation of the equivalent degrees of freedom of smoothing splines: A spectral factorization approach

Smoothing splines are commonly used to reconstruct an unknown continuous function given n discrete noisy samples. In the tuning of the regularization parameter, which controls the balance between smoothness and datafit, the most computerintensive part is the evaluation of the socalled ''equivalent degrees of freedom'' (EDOF) as a function of the regularization parameter. In the paper a closedform expression of the asymptotic (as n goes to infinity) EDOF is obtained for the case of equally spaced data. The derivation is based on the reformulation of the spline smoothing problem as a Bayesian estimation prob lem. Statespace methods, Kalman filtering, and spectral factorization techniques are used to show that the asymp totic EDOF can be obtained as the variance of a suitably defined stationary process. As a byproduct of the main result, it is found that the asymptotic EDOF depend on the cube of the sampling interval.