Asymmetric Cross-Correlation in Multivariate Spatial Stochastic Processes: A Primer

TL;DR

The paper identifies asymmetric cross-covariance as a fundamental feature of multivariate spatial processes and analyzes how the joint covariance structure, particularly the off-diagonal blocks, captures this asymmetry. It reviews unconditional approaches (intrinsic correlation, kernel convolution, and multivariate Matérn) and conditional approaches (Mardia's and Cressie's) for building the joint structure, highlighting each method's capacity to represent asymmetry, and introduces a shift parameter to induce asymmetry in kernel-based and Matérn constructions. A 1D simulation demonstrates that incorporating asymmetry via a shift improves predictive accuracy, underscoring the practical benefits of accommodating asymmetric cross-covariance. The discussion calls for computationally efficient, asymmetric-aware models suitable for large-scale multivariate spatial data across domains such as climate, pandemics, air quality, and socio-economic metrics.

Abstract

Multivariate spatial phenomena are ubiquitous, spanning domains such as climate, pandemics, air quality, and social economy. Cross-correlation between different quantities of interest at different locations is asymmetric in general. This paper provides the visualization, structure, and properties of asymmetric cross-correlation as well as symmetric auto-correlation. It reviews mainstream multivariate spatial models and analyzes their capability to accommodate asymmetric cross-correlation. It also illustrates the difference in model accuracy with and without asymmetric accommodation using a 1D simulated example.

Figures (4)

• Figure 1: Empirical same-component auto-correlation matrix plots for PM2.5 across four longitude strips $[-180^{\circ}, -90^{\circ})$, $[-90^{\circ}, 0^{\circ})$, $[0^{\circ}, 90^{\circ})$, $[90^{\circ}, 180^{\circ}]$. All symmetric about y = x.
• Figure 2: Empirical cross-correlation matrix plots for PM2.5 and Sea Salt across four longitude strips $[-180^{\circ}, -90^{\circ})$, $[-90^{\circ}, 0^{\circ})$, $[0^{\circ}, 90^{\circ})$, $[90^{\circ}, 180^{\circ}]$. All asymmetric.
• Figure 3: Simulated joint covariance matrix $\hbox{$\Sigma$}_{6*40 \times 6*40}$ for six components spanning 40 1D locations. All main diagonal blocks are symmetric, while each off-diagonal block is asymmetric. $\hbox{$\Sigma$}_{6*40 \times 6*40}$ must be positive definite and symmetric, therefore, the asymmetric off-diagonal blocks are symmetric across the main diagonal blocks.
• Figure 4: Prediction results for the first 50 locations of the first field, given the noisy observations from the remaining locations in the first field and the 400 observations from the other two fields.