Abstract
We develop an approach to the spectral estimation that has been advocated by [A. Ferrante et al. , “Time and spectral domain relative entropy: A new approach to multivariate spectral estimation,” IEEE Trans. Autom. Control , vol. 57, no. 10, pp. 2561–2575, Oct. 2012.] and, in the context of the scalar-valued covariance extension problem, by [P. Enqvist and J. Karlsson, “Minimal itakura-saito distance and covariance interpolation,” in Proc. 47th IEEE Conf. Decision Control , 2008, pp. 137–142]. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou, and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In this paper, we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.
Highlights
Consider a stationary, vector-valued, discrete-time, zero-mean, Gaussian stochastic process {y(t) | t ∈ Z}, where y(t) ∈ Rm, and Z, R are the sets of integers and reals, respectively
Following [2], we view this problem in a largedeviations framework where a prior law Q is available, and where this law corresponds to a power spectral density Ψ with finite entropy rate
The moment constraints were cast in the form of the state covariance of an input-to-state filter
Summary
Vector-valued, discrete-time, zero-mean, Gaussian stochastic process {y(t) | t ∈ Z}, where y(t) ∈ Rm, and Z, R are the sets of integers and reals, respectively. We denote the corresponding probability law (on sample paths of the process) by P [1, Chapter 1] and the power spectral density, which we assume exists, by Φ(eiθ), θ ∈ [0, 2π). We assume that the stochastic process is nondeterministic in that the entropy rate is finite, π log det Φ(eiθ)dθ < ∞. Following [2], we view this problem in a largedeviations framework where a prior law Q is available, and where this law corresponds to a power spectral density Ψ with finite entropy rate. The KL divergence between the two laws is precisely the Itakura-Saito distance between the corresponding power spectra, which was considered in [3] for the special case of covariance extension for scalar-valued time series
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