Regularized spectrum estimation using stable spline kernels

• Published in: Automatica (Oxford) Vol. 49; no. 11; pp. 3199 - 3209
• Main Authors: Bottegal, Giulio, Pillonetto, Gianluigi
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.11.2013; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Automation; Computer science; control theory; systems; Control theory. Systems; Data processing. List processing. Character string processing; Exact sciences and technology; Hilbert space; Kalman smoothing; Kernel-based methods; Kernels; Machine learning; Mathematical analysis; Mathematical models; Mathematics; Memory organisation. Data processing; Modelling and identification; Nonparametric estimation; Probability and statistics; Probability theory and stochastic processes; Reproducing kernel Hilbert spaces; Sciences and techniques of general use; Software; Splines; Stochastic processes; System identification; Theoretical computing; Tuning; Nonparametric estimation; Reproducing kernel Hilbert spaces; Kalman smoothing; System identification; Kernel-based methods; Machine learning; Second moment; Zero of function; Random excitation; Kalman filter; Non parametric estimation; Cross validation; Stochastic process; Spline approximation; Model matching; Observer; Variational calculus; Correlation function; Hilbert space; Stationary process; Transfer function; Oracle; Computational complexity; Standards; Kernel method; Smoothing; Learning (artificial intelligence); Artificial intelligence
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2013.08.010

This paper presents a new regularized kernel-based approach for the estimation of the second order moments of stationary stochastic processes. The proposed estimator is defined by a Tikhonov-type variational problem. It contains few unknown parameters which can be estimated by cross validation solving a sequence of problems whose computational complexity scales linearly with the number of noisy moments (derived from the samples of the process). The correlation functions are assumed to be summable and the hypothesis space is a reproducing kernel Hilbert space induced by the recently introduced stable spline kernel. In this way, information on the decay to zero of the functions to be reconstructed is incorporated in the estimation process. An application to the identification of transfer functions in the case of white noise as input is also presented. Numerical simulations show that the proposed method compares favorably with respect to standard nonparametric estimation algorithms that exploit an oracle-type tuning of the parameters.