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

Wang Yinbu, Xu Yong

TL;DR

This work introduces a geometry-driven test for wide-sense stationarity (WSS) that leverages the cylindrical structure of the covariance surface $r(s,t)$ under WSS, asserting $r_s + r_t = 0$. By constructing a locally cylindrical approximation via patchwise division and estimating derivatives with local polynomial regression, the authors form $J = r_s + r_t$ and test whether $J=0$ in time-resolved segments. The approach is model-free and applicable to stochastic dynamical systems, with rigorous asymptotic guarantees for the estimator $\hat J$ and demonstrated effectiveness on SDOF and stochastic Duffing oscillators, including comparisons to the Dette DPV measure. The method provides interpretable diagnostics of nonstationarity directly on the covariance surface and offers a path toward a local spectral/radar-like view of evolving stationarity in complex stochastic 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.

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

TL;DR

This work introduces a geometry-driven test for wide-sense stationarity (WSS) that leverages the cylindrical structure of the covariance surface under WSS, asserting . By constructing a locally cylindrical approximation via patchwise division and estimating derivatives with local polynomial regression, the authors form and test whether in time-resolved segments. The approach is model-free and applicable to stochastic dynamical systems, with rigorous asymptotic guarantees for the estimator and demonstrated effectiveness on SDOF and stochastic Duffing oscillators, including comparisons to the Dette DPV measure. The method provides interpretable diagnostics of nonstationarity directly on the covariance surface and offers a path toward a local spectral/radar-like view of evolving stationarity in complex stochastic 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 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.
Paper Structure (13 sections, 8 theorems, 101 equations, 11 figures, 2 tables)

This paper contains 13 sections, 8 theorems, 101 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Geometric Shape of the Covariance Function
  3. Geometric Shape
  4. Local Cylindrification
  5. WSS Testing
  6. Local Polynomial Regression
  7. Partial derivative estimation
  8. Asymptotic theory
  9. Simulation
  10. SDOF
  11. Duffing Oscillator
  12. Comparison
  13. Conclusion

Key Result

Lemma 2.1

The process $X_t$ is WSS on a connected open set $T$ with zero mean if and only if its covariance $r(s,t)\in C^1(T\times T)$ and Here

Figures (11)

  • Figure 1: A surface $M$ generated by straight rulings with the same direction.
  • Figure 2: Local cylindrification on a surface patch $\Omega$ with parallel rulings and a unique ruling direction.
  • Figure 3: A sample path of SDOF Response.
  • Figure 4: Time series of the statistic $\hat{J}$.
  • Figure 5: Time series of $\hat{J}$ over $[0,30]$ seconds for different sample sizes.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.6
  • proof
  • ...and 6 more