Highly Multivariate Large-scale Spatial Stochastic Processes – A Cross-Markov Random Field Approach

Key challenges in the analysis of highly multivariate large-scale spatial stochastic processes, where both the number of components (p) and spatial locations (n) can be large, include achieving maximal sparsity in the joint precision matrix, ensuring efficient computational cost for its generation, accommodating asymmetric cross-covariance in the joint covariance matrix, and delivering scientific interpretability. We propose a cross-Markov Random Field model class, consisting of a mixed spatial graphical model framework and cross-Markov Random Field theory, to collectively address these challenges in one unified framework. We demonstrate with 1D simulated comparative studies and 2D real-world data.

auto-neighbourhood

cross-neighbourhood

cross-Markov Random Field

doubly conditional independence

mixed spatial graph

spatial stochastic processes 

Section on Statistics and the Environment