Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space–time fields on arbitrary manifolds, and practical computational considerations.

高斯过程和随机场有着悠久的历史，涵盖了表示空间和时空依赖结构的多种方法，例如协方差函数、谱表示、再现核希尔伯特空间和基于图的模型。本文描述了通过希尔伯特空间投影推广 Matérn 协方差模型的随机偏微分方程方法如何与这些方法中的几种连接，每种连接在不同情况下都有用。除了对主要思想的概述外，还讨论了一些重要的扩展、理论、应用和其他最新发展。这些方法包括马尔可夫和非马尔可夫模型、非高斯随机场、非平稳场和任意流形上的时空场，