A Wide-Sense Stationarity Test Based on the Geometric Structure of Covariance

TL;DR

This paper introduces a novel geometric approach to testing wide-sense stationarity in stochastic processes by analyzing the covariance surface, applicable to various dynamical systems and capable of detecting non-stationarity.

Contribution

It proposes a new WSS test based on covariance surface geometry that does not assume stationarity and is applicable to general stochastic systems.

Findings

Method is numerically stable.

Can detect departures from WSS.

Applicable to complex dynamical systems.

Abstract

This paper presents a test for wide-sense stationarity (WSS) based on the geometry of the covariance function. We estimate local patches of the covariance surface and then check whether the directional derivative in the $(1,1,0)$ direction is zero on each patch. The method only requires the covariance function to be locally smooth and does not assume stationarity in advance. It can be applied to general stochastic dynamical systems and provides a time-resolved view. We apply the test method to an SDOF system and to a stochastic Duffing oscillator. These examples show that the method is numerically stable and can detect departures from WSS in practice.