Fast spline smoothing via spectral factorization concepts

When tuning the smoothness parameter of nonparametric regression splines, t he evaluation of the so-called degrees of freedom is one of the most comput er-intensive tasks. In the paper, a closed-form expression of the degrees o f freedom is obtained for the case of cubic splines and equally spaced data when the number of data tends to infinity. State-space methods, Kalman fil tering and spectral factorization techniques are used to prove that the asy mptotic degrees of freedom are equal to the variance of a suitably defined stationary process. The closed-form expression opens the way to fast spline smoothing algorithms whose computational complexity is about one-half of s tandard methods (or even one-fourth under further approximations). (C) 2000 Elsevier Science Ltd. All rights reserved.