In the task of predicting spatio-temporal fields in environmental science, introducing models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest in spatial statistics. The size of space-time datasets calls for new numerical methods to efficiently process them. The SPDE (Stochastic Partial Differential Equation) approach has proven to be effective for the estimation and the prediction in a spatial context. We present here the advection-diffusion SPDE with first order derivative in time to enlarge the SPDE family to the space-time context. By varying the coefficients of the differential operators, the approach allows to define a large class of non-separable spatio-temporal models. A Gaussian Markov random field approximation of the solution of the SPDE is built by discretizing the temporal derivative with a finite difference method (implicit Euler) and by solving the purely spatial SPDE with a finite element method (continuous Galerkin) at each time step. The ''Streamline Diffusion'' stabilization technique is introduced when the advection term dominates the diffusion term. Computationally efficient methods are proposed to estimate the parameters of the SPDE and to predict the spatio-temporal field by kriging. The approach is applied to a solar radiation dataset. Its advantages and limitations are discussed.
翻译:在环境科学中预测时空空间领域的任务中,引入由数字效率基本现象的物理原理所启发的模型,对空间统计的兴趣越来越大。空间时间数据集的规模要求采用新的数字方法来有效处理它们。SPDE(随机部分差异化)方法已证明对空间范围内的估算和预测是有效的。我们在这里展示了通过第一顺序衍生物来及时将SPDE大家庭扩大至时空背景的对流扩散SPDE。通过差异操作者的系数不同,该方法可以定义大量非可分离的时空模型。Gausian Markov 随机匹配SPDE解决方案的字段,其方法是将时间衍生物与有限的差异方法(模糊的 Euler)分离,并在每个时间步骤中用有限的元素方法(连续的 Galerkin)解决纯空间SPDE。“Sreline Diffl” 稳定技术是“Sparine Difficulation”技术,当对高效的辐射参数预测方法加以应用时,通过对空间数据定义的对空间参数进行精确性估算,从而将数据应用到空间预测方法。