Divergence Families for THREE-like Multivariate Spectral Estimation

Abstract:  The Tunable High Resolution Estimator (THREE) and its extensions represent a modern tool to estimate the spectral density of a multivariate stochastic process.  The former has  been introduced by Christopher Byrnes, Tryphon Georgiou and Anders Lindquist. 

The appealing feature of those methods is that they lead to a convex optimization problem whose solution is a spectral density having bounded McMillan degree. 

A THREE-like method may be outlined as follows. A finite length sequence extracted from a realization of a stochastic process, say y, is fed to a bank of filters. The estimated output covariance matrix is then used to extract information on the process by considering the set of spectral densities matching such output covariance. Therefore, the estimate of the spectral density of y is chosen in this set. This task is accomplished by solving a spectrum approximation problem whose solution minimizes a divergence index with respect to an a priori spectral density. It turns out that the features of the solution highly depends  on the chosen divergence, in particular the upper bound on the complexity of the solution, in terms of McMillan degree. In this talk, we analyze three different (families of) solutions obtained by using the Alpha, Beta and Tau divergence families.