PII: S0005-1098(00)00100-X Copyright © 2000 Elsevier Science B.V. All rights reserved.
Fast spline smoothing via spectral factorization concepts*1 Giuseppe De Nicolao1, , , , a, Giancarlo Ferrari-Trecate1, , , a and Giovanni Sparacino2, , , b a Dipartimento di Informatica e Sistemistica, Universitą degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italyb Dipartimento di Elettronica e Informatica, Universitą di Padova, Via Gradenigo 6/A, 35122 Padova, Italy Received 23 June 1999; revised 31 January 2000; accepted 6 April 2000. Available online 30 August 2000.
Giuseppe De Nicolao1, , , , a, Giancarlo Ferrari-Trecate1, , , a and Giovanni Sparacino2, , , b
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).
Author Keywords: Spline smoothing; Bayes estimation; Kalman filters; Stochastic processes; Spectral factorization
*1 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Tits under the direction of Editor T. Basar.
1 Supported by MURST Project "Identification and control of industrial systems".
2 Supported by the NIH grant RR11095 "Input Estimation of Biological Systems by Deconvolution".
Corresponding author. Tel.: +39-382-505484; fax: +39-382-505373; email: denicolao@conpro.unipv.it
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