Understanding and predicting environmental phenomena often requires the construction of spatio-temporal statistical models, which are typically Gaussian processes. A common assumption made on Gaussian processes is that of covariance stationarity, which is unrealistic in many geophysical applications. In this article, we introduce a deep-learning-inspired approach to construct descriptive nonstationary spatio-temporal models by modeling stationary processes on warped spatio-temporal domains. The warping functions we use are constructed using several simple injective warping units which, when combined through composition, can induce complex warpings. A stationary spatio-temporal covariance function on the warped domain induces covariance nonstationarity on the original domain. Sparse linear algebraic methods are used to reduce the computational complexity when fitting the model in a big data setting. We show that our proposed nonstationary spatio-temporal model can capture covariance nonstationarity in both space and time, and provide better probabilistic predictions than conventional stationary models in both simulation studies and on a real-world data set.

翻译：理解和预测环境现象通常需要构建时空统计模型，这些模型通常是高斯过程。高斯过程的一个常见假设是协方差平稳性，但这在许多地球物理应用中并不现实。本文提出了一种受深度学习启发的方法，通过在扭曲的时空域上对平稳过程进行建模，从而构建描述性非平稳时空模型。我们使用的扭曲函数由多个简单的单射扭曲单元构成，这些单元通过复合组合可以产生复杂的扭曲。扭曲域上的平稳时空协方差函数会在原始域上诱导出协方差非平稳性。采用稀疏线性代数方法可降低大数据场景下模型拟合的计算复杂度。研究表明，我们提出的非平稳时空模型能够捕捉空间和时间上的协方差非平稳性，并且在模拟研究和真实数据集上均能提供比传统平稳模型更优的概率预测。