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广义平稳性（WSS）检验方法。我们首先估计协方差曲面的局部区域，随后检验每个区域上沿 $(1,1,0)$ 方向的协方差方向导数是否为零。该方法仅要求协方差函数局部光滑，无需预先假设平稳性，可应用于一般随机动力系统并提供时间分辨视角。我们将该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性良好，并能有效检测实际系统中偏离广义平稳性的现象。