The Matérn covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based'' extension of the spatial Matérn covariance model to a spatio-temporal non-separable covariance model that allows fractional smoothnesses in space and in time. The model is described in terms of a space-time fractional stochastic partial differential equation, but currently proposed computational approaches have strong restrictions on the possible smoothnesses in time. We propose a discretization method based on rational approximations in time to handle arbitrary smoothnesses, which leads to a vector autoregressive moving average process (VARMA). We prove that the covariance function of the approximation converges pointwise, determine explicit convergence rates as a function of spatial and temporal resolutions and the accuracy of the rational approximation, and conduct numerical verification to demonstrate small pointwise error for low orders of the VARMA process. Through a simulation study, we demonstrate that the parameters can be estimated back and that correctly specifying the temporal smoothness is especially important for forecasting. The approach is illustrated for three months of daily mean temperatures in mainland France.

翻译：马特恩协方差模型在空间建模中普遍存在，但在时空建模中尚无默认选择。本文考虑近期提出的基于扩散的时空非分离协方差模型——该模型对空间马特恩协方差模型进行扩展，允许在空间和时间维度上具有分数阶平滑性。该模型通过时空分数阶随机偏微分方程描述，但现有计算方法对时间方向的平滑性参数存在严格限制。我们提出一种基于时间方向有理近似的离散化方法，可处理任意平滑性参数，由此得到向量自回归滑动平均过程(VARMA)。我们证明了近似协方差函数的逐点收敛性，确定了收敛速率与空间/时间分辨率及有理近似精度的显式关系，并通过数值验证表明低阶VARMA过程具有较小的逐点误差。通过仿真研究，我们证明了参数可以被准确估计，且正确设定时间平滑性对预测尤为关键。该方法已应用于法国本土三个月日平均气温数据。