Time and spectral domain relative entropy: A new approach to multivariate spectral estimation

TL;DR

This paper introduces a novel multivariate spectral estimation method based on spectral relative entropy, connecting time and spectral domain rates, and demonstrates its effectiveness especially with short data records.

Contribution

It extends the THREE approach to multivariate processes using a new spectral distance, improving complexity bounds and providing a convergent algorithm.

Findings

Effective in multivariate spectral estimation with short data records

Improves complexity bounds over previous methods

Demonstrates convergence of the proposed algorithm

Abstract

The concept of spectral relative entropy rate is introduced for jointly stationary Gaussian processes. Using classical information-theoretic results, we establish a remarkable connection between time and spectral domain relative entropy rates. This naturally leads to a new spectral estimation technique where a multivariate version of the Itakura-Saito distance is employed}. It may be viewed as an extension of the approach, called THREE, introduced by Byrnes, Georgiou and Lindquist in 2000 which, in turn, followed in the footsteps of the Burg-Jaynes Maximum Entropy Method. Spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate. The corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework. Indeed, it is equal to the one featured by THREE in the scalar case. The solution is computed via a globally convergent matricial Newton-type algorithm. Simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks, especially in the case of short data records.