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May 2015 Cross-Covariance Functions for Multivariate Geostatistics
Marc G. Genton, William Kleiber
Statist. Sci. 30(2): 147-163 (May 2015). DOI: 10.1214/14-STS487


Continuously indexed datasets with multiple variables have become ubiquitous in the geophysical, ecological, environmental and climate sciences, and pose substantial analysis challenges to scientists and statisticians. For many years, scientists developed models that aimed at capturing the spatial behavior for an individual process; only within the last few decades has it become commonplace to model multiple processes jointly. The key difficulty is in specifying the cross-covariance function, that is, the function responsible for the relationship between distinct variables. Indeed, these cross-covariance functions must be chosen to be consistent with marginal covariance functions in such a way that the second-order structure always yields a nonnegative definite covariance matrix. We review the main approaches to building cross-covariance models, including the linear model of coregionalization, convolution methods, the multivariate Matérn and nonstationary and space–time extensions of these among others. We additionally cover specialized constructions, including those designed for asymmetry, compact support and spherical domains, with a review of physics-constrained models. We illustrate select models on a bivariate regional climate model output example for temperature and pressure, along with a bivariate minimum and maximum temperature observational dataset; we compare models by likelihood value as well as via cross-validation co-kriging studies. The article closes with a discussion of unsolved problems.


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Marc G. Genton. William Kleiber. "Cross-Covariance Functions for Multivariate Geostatistics." Statist. Sci. 30 (2) 147 - 163, May 2015.


Published: May 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1332.86010
MathSciNet: MR3353096
Digital Object Identifier: 10.1214/14-STS487

Keywords: asymmetry , Co-kriging , multivariate random fields , nonstationarity , separability , smoothness , spatial statistics , symmetry

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.30 • No. 2 • May 2015
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