Gaussian random fields with Mat\'ern covariance functions are popular models in spatial statistics and machine learning. In this work, we develop a spatio-temporal extension of the Gaussian Mat\'ern fields formulated as solutions to a stochastic partial differential equation. The spatially stationary subset of the models have marginal spatial Mat\'ern covariances, and the model also extends to Whittle-Mat\'ern fields on curved manifolds, and to more general non-stationary fields. In addition to the parameters of the spatial dependence (variance, smoothness, and practical correlation range) it additionally has parameters controlling the practical correlation range in time, the smoothness in time, and the type of non-separability of the spatio-temporal covariance. Through the separability parameter, the model also allows for separable covariance functions. We provide a sparse representation based on a finite element approximation, that is well suited for statistical inference and which is implemented in the R-INLA software. The flexibility of the model is illustrated in an application to spatio-temporal modeling of global temperature data.

翻译：具有Matérn协方差函数的高斯随机场是空间统计和机器学习中的常用模型。本文提出了一种高斯Matérn场的时空扩展模型，该模型被表述为随机偏微分方程的解。该模型的空间平稳子集具有边际空间Matérn协方差，并进一步扩展到弯曲流形上的Whittle-Matérn场及更一般的非平稳场。除空间依赖参数（方差、平滑度和实际相关范围）外，该模型还包含控制时间实际相关范围、时间平滑度以及时空协方差非可分离性类型的参数。通过可分离性参数，该模型还可实现可分离协方差函数。我们基于有限元近似提供了稀疏表示，该表示非常适合统计推断，并已在R-INLA软件中实现。该模型的灵活性通过全球温度数据的时空建模应用得到展示。