Cross-Covariance Functions for Multivariate Geostatistics

TL;DR

This paper reviews methods for modeling relationships between multiple spatial variables using cross-covariance functions, addressing challenges in ensuring valid multivariate spatial models and illustrating with climate data examples.

Contribution

It provides a comprehensive review of approaches to constructing cross-covariance functions, including recent extensions and specialized models, with practical applications and model comparisons.

Findings

Models compared by likelihood and cross-validation.

Multivariate models improve joint spatial predictions.

Various constructions handle nonstationarity and domain-specific features.

Abstract

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 Mat\'{e}rn 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.