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.

翻译：本文提出了一种基于协方差函数几何结构的宽平稳性检验方法。我们通过估计协方差曲面的局部区域，并检验每个区域上沿 $(1,1,0)$ 方向的导数值是否为零。该方法仅要求协方差函数局部光滑，无需预先假设平稳性，可应用于一般的随机动力系统，并提供时间分辨的视角。我们将此检验方法应用于单自由度系统及随机Duffing振子，算例表明该方法数值稳定，且能有效检测实际系统中对宽平稳性的偏离。