Space-Time Covariance Structures and Models

Abstract

In recent years, interest has grown in modeling spatio-temporal data generated from monitoring networks, satellite imaging, and climate models. Under Gaussianity, the covariance function is core to spatio-temporal modeling, inference, and prediction. In this article, we review the various space-time covariance structures in which simplified assumptions, such as separability and full symmetry, are made to facilitate computation, and associated tests intended to validate these structures. We also review recent developments on constructing space-time covariance models, which can be separable or nonseparable, fully symmetric or asymmetric, stationary or nonstationary, univariate or multivariate, and in Euclidean spaces or on the sphere. We visualize some of the structures and models with visuanimations. Finally, we discuss inference for fitting space-time covariance models and describe a case study based on a new wind-speed data set. Expected final online publication date for the Annual Review of Statistics, Volume 8 is March 8, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Modeling and Predicting Spatio-temporal Dynamics of PM$_{2.5}$ Concentrations Through Time-evolving Covariance Models

“…The particular restrictions C(h, 0) and C(0, u) represent purely spatial and purely temporal covariance functions, respectively. The existing literature on stationary spatio-temporal models provides practitioners with numerous alternatives, and those are comprehensively summarized in review papers by Kyriakidis and Journel (1999), Gneiting et al (2006) and Chen et al (2021). A rudimentary approach to build a valid spatio-temporal covariance function is to impose separabilty, in which C(h, u) can be decomposed into purely spatial and purely temporal covariance function.…”

Covariance Models for Multivariate Random Fields resulting from Pseudo Cross-Variograms

“…This intimate connection adds another dimension to the range of applications of pseudo cross-variograms, that is, the construction of valid covariance models for multivariate random fields. In this regard, pseudo crossvariograms have already been used in Dörr and Schlather (2021), Allard et al (2022) and Porcu et al (2022) to propose several extensions of Gneiting's popular univariate space-time covariance model (Gneiting, 2002b), thereby meeting one of the requests of Chen et al (2021) for flexible space-time cross-covariance models.…”

Improving the accuracy of brain activation maps in the group-level analysis of fMRI data utilizing spatiotemporal Gaussian process model

“…Failure to establish this hypothesis leads to misspecification of the mean or covariance structure. [24]. Aston et al [25] suggested a method for separability hypothesis testing in high-dimensional and hyper-surfaces.…”

“…Aston et al [25] suggested a method for separability hypothesis testing in high-dimensional and hyper-surfaces. However, when separability is considered, it does not allow any interaction between time and space and leads to discontinuity in correlations [24]. As an example of using separable structure modeling, Lorenzi et al [26] considered a spatiotemporal structure using the kernel convolution of a white noise Gaussian process.…”