The Matérn Model: A Journey Through Statistics, Numerical Analysis and Machine Learning

The Mat & eacute;rn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Mat & eacute;rn model has been exploited in disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article, we take a Mat & eacute;rn-based journey across these disciplines. First, we reflect on the importance of the Mat & eacute;rn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Mat & eacute;rn model has been influential. Then, we position the Mat & eacute;rn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Mat & eacute;rn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.