# Asymmetric Matrix-Valued Covariances for Multivariate Random Fields on Spheres

## TL;DR

This paper introduces a novel approach for constructing asymmetric matrix-valued covariance functions on spheres, improving the modeling of multivariate spatial data over large geographic areas.

## Contribution

It proposes a rotation-based method to generate asymmetric covariances on spheres, addressing the limitations of symmetric models in geostatistics.

## Key findings

- Enhanced predictive performance over symmetric models
- Effective in real data analysis and simulations
- Addresses the gap in asymmetric covariance modeling on spheres

## Abstract

Matrix-valued covariance functions are crucial to geostatistical modeling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overlay restrictive and has been considered as unrealistic for most of real data applications. Despite of that, the literature on asymmetric covariance functions has been very sparse. In particular, there is some work related to asymmetric covariances on Euclidean spaces, depending on the Euclidean distance. However, for data collected over large portions of planet Earth, the most natural spatial domain is a sphere, with the corresponding geodesic distance being the natural metric. In this work, we propose a strategy based on spatial rotations to generate asymmetric covariances for multivariate random fields on the $d$-dimensional unit sphere. We illustrate through simulations as well as real data analysis that our proposal allows to achieve improvements in the predictive performance in comparison to the symmetric counterpart.