Correlation and smoothness are terms used to describe a wide variety of random quantities. In time, space, and many other domains, they both imply the same idea: quantities that occur closer together are more similar than those further apart. Two popular statistical models that represent this idea are basis-penalty smoothers (Wood in Texts in statistical science, CRC Press, Boca Raton, 2017) and stochastic partial differential equations (SPDEs) (Lindgren et al. in J R Stat Soc Series B (Stat Methodol) 73(4):423–498, 2011). In this paper, we discuss how the SPDE can be interpreted as a smoothing penalty and can be fitted using the R package mgcv , allowing practitioners with existing knowledge of smoothing penalties to better understand the implementation and theory behind the SPDE approach. Supplementary materials accompanying this paper appear online.

中文翻译：

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理解平滑的随机偏微分方程方法

相关性和平滑度是用于描述各种随机量的术语。在时间、空间和许多其他领域，它们都暗示着相同的想法：发生得更近的数量比距离更远的数量更相似。代表这个想法的两个流行的统计模型是基础惩罚平滑器（Wood in Texts in Statistics Science，CRC Press，Boca Raton，2017）和随机偏微分方程（SPDE）（Lindgren 等人在 JR Stat Soc 系列 B（Stat Methodol) 73(4):423–498, 2011)。在本文中，我们讨论了如何将 SPDE 解释为平滑惩罚并使用 R 包 mgcv 进行拟合，从而使具有平滑惩罚现有知识的从业者能够更好地理解 SPDE 方法背后的实现和理论。