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无处不在，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空间马特恩协方差模型向时空非可分离协方差模型的扩展，该模型允许空间和时间上的分数阶平滑度。该模型通过时空分数阶随机偏微分方程描述，但当前提出的计算方法对时间上的可能平滑度存在严格限制。我们提出一种基于时间有理近似的离散化方法以处理任意平滑度，该方法导出向量自回归移动平均过程（VARMA）。我们证明该近似协方差函数逐点收敛，确定显式收敛速率（作为空间和时间分辨率以及有理近似精度的函数），并通过数值验证展示低阶VARMA过程的极小逐点误差。通过模拟研究，我们证明参数可被反估计，且正确指定时间平滑度对预测尤为重要。该方法已应用于法国本土三个月日均气温的实测数据。