Multivariate Spatial Covariance Models: A Conditional Approach

TL;DR

This paper introduces a conditional approach for constructing multivariate spatial covariance models that ensures validity and simplifies model comparison, demonstrated through temperature and pressure data analysis.

Contribution

The paper presents a new conditional modeling framework for multivariate spatial covariance that simplifies validity checks and model comparison based on network structures.

Findings

The approach ensures nonnegative-definite covariance matrices.

Application to temperature and pressure data shows asymmetric variable roles.

Network structure aids in model selection and causal inference.

Abstract

Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables at any sets of locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model must be nonnegative-definite. In this article, we develop a conditional approach for spatial-model construction whose validity conditions are easy to check. We start with bivariate spatial covariance models and go on to demonstrate the approach's connection to multivariate models defined by networks of spatial variables. In some circumstances, such as modelling respiratory illness conditional on air pollution, the direction of conditional dependence is clear. When it is not, the two directional models can be compared. More generally, the graph structure of the network reduces the number of possible models to compare. Model selection then amounts to finding possible causative links in the network. We demonstrate our conditional approach on surface temperature and pressure data, where the role of the two variables is seen to be asymmetric.