30 Years of space–time covariance functions. (20th May 2020) - Porcu, Emilio; Furrer, Reinhard; Nychka, Douglas
Abstract: In this article, we provide a comprehensive review of space–time covariance functions. As for the spatial domain, we focus on either the d ‐dimensional Euclidean space or on the unit d ‐dimensional sphere. We start by providing background information about (spatial) covariance functions and their properties along with different types of covariance functions. While we focus primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships. We discuss properties of space–time covariance functions along with the relevant results associated with spectral representations. Special attention is given to the Gneiting class of covariance functions, which has been especially popular in space–time geostatistical modeling. We then discuss some techniques that are useful for constructing new classes of space–time covariance functions. Separate treatment is reserved for spectral models, as well as to what are termed models with special features . We also discuss the problem of estimation of parametric classes of space–time covariance functions. An outlook concludes the paper. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Abstract : A separable covariance function (left) and a space‐time covariance function with dynamical compact support.