G. De Nicolao, G. Ferrari-Trecate, and G. Sparacino. Fast spline smoothing via spectral factorization concepts. Automatica , 36(11):1733--1739, 2000. Link to Automatica
When tuning the smoothness parameter of nonparametricregression splines, the evaluation ofthe so-called degreesof freedom is one of the most computer-intensive tasks.In the paper, aclosed-form expression of the degrees of freedom is obtained for the case ofcubic splines and equally spaced data when the number of data tends to infinity.State-space methods, Kalman filtering and spectral factorization techniques areused to prove that the asymptotic degrees of freedom are equal to the varianceof a suitably defined stationary process. The closed-form expression opens thewayto fast spline smoothing algorithms whose computational complexity is aboutone half of standard methods (or even one fourth under further approximations).