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

It is common to assess the “memory strength” of a stationary process by 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). Minimizing this measure is then 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.