Brief Fast spline smoothing via spectral factorization concepts

Authors:
Giuseppe De Nicolao;Giancarlo Ferrari-Trecate;Giovanni Sparacino
Affiliations:
Dipartimento di Informatica e Sistemistica, Universití degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy;Dipartimento di Informatica e Sistemistica, Universití degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy;Dipartimento di Elettronica e Informatica, Universití di Padova, Via Gradenigo 6/A, 35122 Padova, Italy
Venue:
Automatica (Journal of IFAC)
Year:
2000

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Abstract

When tuning the smoothness parameter of nonparametric regression splines, the evaluation of the so-called degrees of freedom is one of the most computer-intensive tasks. In the paper, a closed-form expression of the degrees of 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 filtering and spectral factorization techniques are used to prove that the asymptotic 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 standard methods (or even one-fourth under further approximations).