G. Ferrari-Trecate and G. De Nicolao. Computing the equivalent number of parameters of fixed-interval smoothers. Technical Report AUT01-14, Automatic Control Laboratory, ETH Zurich, 2001. http://control.ethz.ch/
The problem of reconstructing an unknown signal from $n$ noisysamples can be addressed by means of nonparametric estimationtechniques such as Tikhonov regularization, Bayesian regressionand state-space fixed-interval smoothing. The practical use ofthese approaches calls for the tuning of a regularizationparameter that controls the amount of smoothing they introduce.The leading tuning criteria, including Generalized CrossValidation and Maximum Likelihood, involve the repeatedcomputation of the so-called equivalent number of parameters,a normalized measure of the flexibility of thenonparametric estimator. The paper develops new state-spaceformulas for the computation of the equivalent number of parameters in $O(n)$ operations.The results are specialized to the case of uniform samplingyielding closed-form expressions of the equivalent number of parameters for both linear splinesand first-order deconvolution.