Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Matérn Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

翻译：本征高斯场在统计学的许多领域中被用作空间或时空依赖性的模型，或作为潜变量的先验分布。然而，现有文献存在两个主要空白：首先，现有本征模型的数量和灵活性非常有限；其次，针对本征场的理论、快速推断方法和软件目前尚不完善。我们通过引入一类新的、灵活的本征Whittle--Matérn高斯随机场来解决这些挑战，该随机场由随机偏微分方程（SPDE）的解获得。利用有限元近似所产生的稀疏性，我们为这些模型开发了快速的估计和模拟方法。我们证明了这种本征SPDE方法在空间克里金插值外推这一重要任务中的优势。利用本征场与空间极值过程的联系，我们将理论推广至Brown--Resnick过程的SPDE方法，用于空间极端事件的稀疏建模。这一新范式为在空前维度上进行高效推断铺平了道路。为展示新方法的广泛适用性，我们将其应用于两个截然不同的领域：肾功能数据的纵向研究，以及利用高分辨率海表温度数据对海洋热浪的建模。