Likelihood Analysis of Power Spectra and Generalized Moment Problems

TL;DR

This paper introduces a new fast algorithm for spectral estimation that finds power spectra consistent with given moments, minimizing relative entropy, and generalizes classical maximum entropy methods.

Contribution

It presents a novel fast algorithm for spectral estimation with Gaussian priors and provides closed-form solutions for specific prior structures.

Findings

Developed a new efficient algorithm for spectral estimation.

Extended the maximum entropy framework to more general Gaussian priors.

Provided closed-form solutions for certain structured priors.

Abstract

We develop an approach to spectral estimation that has been advocated by Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance extension problem, by Enqvist and Karlsson. 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 the present 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.