Quantifying and Computing Covariance Uncertainty

TL;DR

This paper develops a method to efficiently bound the uncertainty in covariance functions of stationary signals outside known data points, using convex optimization, with exact results for certain spectral cases.

Contribution

It introduces a finite-dimensional convex approach to bound covariance uncertainty, transforming an infinite-dimensional problem into a computationally feasible one.

Findings

The upper bound on covariance discrepancy can be computed efficiently.

For signals with spectral support on an interval, the bound is exact.

The method enables precise uncertainty quantification in covariance estimation.

Abstract

In this work, we consider the problem of bounding the values of a covariance function corresponding to a continuous-time stationary stochastic process or signal. Specifically, for two signals whose covariance functions agree on a finite discrete set of time-lags, we consider the maximal possible discrepancy of the covariance functions for real-valued time-lags outside this discrete grid. Computing this uncertainty corresponds to solving an infinite dimensional non-convex problem. However, we herein prove that the maximal objective value may be bounded from above by a finite dimensional convex optimization problem, allowing for efficient computation by standard methods. Furthermore, we empirically observe that for the case of signals whose spectra are supported on an interval, this upper bound is sharp, i.e., provides an exact quantification of the covariance uncertainty.