The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field $u$ on $\mathbb{R}^d$ as the solution of an elliptic SPDE $L^\beta u = \mathcal{W}$ where $L$ is a second-order differential operator, $2\beta$ (belongs to natural number starting from 1) is a positive parameter that controls the smoothness of $u$ and $\mathcal{W}$ is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying $\beta>d/4$. Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator $L^{-2\beta}$ of the Gaussian field $u$ by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference.

翻译：随机偏微分方程（SPDE）方法被广泛用于大规模空间数据建模。该方法将定义在 $\mathbb{R}^d$ 上的高斯随机场 $u$ 表示为椭圆型SPDE $L^\beta u = \mathcal{W}$ 的解，其中 $L$ 是二阶微分算子，$2\beta$（属于从1起始的自然数）是控制光滑度的正参数，$\mathcal{W}$ 为高斯白噪声。现有文献提出了若干方法将该方法推广至任意满足 $\beta>d/4$ 的光滑参数。尽管这些方法在模拟一般光滑度的SPDE时效果良好，但它们无法像原始SPDE方法那样提供高斯马尔可夫随机场（GMRF）近似，因此不适用于贝叶斯推断。我们通过提出新方法解决了该问题：采用有限元法结合分数阶幂的有理逼近，对高斯场 $u$ 的协方差算子 $L^{-2\beta}$ 进行近似。由此得到数值稳定的GMRF近似，可与集成嵌套拉普拉斯近似（INLA）方法结合，实现快速贝叶斯推断。我们对该方法进行了严格的收敛性分析，并通过模拟数据验证其精度。最后，通过R包rSPDE结合R-INLA软件对降水数据的全贝叶斯推断应用，展示了该方法及其实现。