Monotonicity of the Trace-Inverse of Covariance Submatrices and Two-Sided Prediction

TL;DR

This paper introduces a new measure based on the normalized trace-inverse of covariance submatrices to characterize the memory strength of stationary processes, linking it to two-sided prediction errors and spectral estimation.

Contribution

It establishes the monotonicity of the trace-inverse sequence, characterizes white processes, and extends the measure to non-stationary processes as an alternative to entropy rate.

Findings

Trace-inverse sequence is monotonically non-decreasing.

The sequence is constant if and only if the process is white.

Proposes a spectral estimation method alternative to Burg's principle.

Abstract

It is common to assess the "memory strength" of a stationary process looking at how fast the normalized log-determinant of its covariance submatrices (i.e., entropy rate) decreases. In this work, we propose an alternative characterization in terms of the normalized trace-inverse of the covariance submatrices. We show that this sequence is monotonically non-decreasing and is constant if and only if the process is white. Furthermore, while the entropy rate is associated with one-sided prediction errors (present from past), the new measure is associated with two-sided prediction errors (present from past and future). This measure can be used as an alternative to Burg's maximum-entropy principle for spectral estimation. We also propose a counterpart for non-stationary processes, by looking at the average trace-inverse of subsets.