Theory and Application of Highly Multivariate High-dimensional Spatial Stochastic Processes

Abstract

The global reanalysis data set produced by the Copernicus Atmosphere Monitoring Service (CAMS) comprises gridded concentration estimates of various pollutants. The complex inter-pollutant relationships across a large spatial domain characterise the data as highly multivariate and spatially high-dimensional (HMHD). Sparsity among variates p and spatial locations n is the key to addressing the HMHD spatial data problems. Without such sparsity, the joint var-covariance matrix Σnp×np and the precision matrix (Σnp×np)^{-1}, where both p and n are large, would be prohibitive to construct and intractable for inference. The thesis proposes a hybrid mixed spatial graphical model framework and novel concepts such as cross-Markov Random Field (cross-MRF) to comprehensively address all aspects of HMHD spatial data features. Specifically, the framework accommodates any customised conditional independence (CI) among any number of p variate fields at the first stage, alleviating the dynamic memory burden associated with Σnp×np construction. Meanwhile, it facilitates parallelled generation of covariance and precision matrix, with the latter's generation order only scaling linearly in p. In the second stage, the thesis demonstrates the multivariate Hammersley-Clifford theorem from a column-wise conditional perspective and unearths the existence of cross-MRF. The link of the mixed spatial graphical framework and the cross-MRF allows for a mixed conditional approach which achieves the sparsest possible representation of the precision matrix via accommodating the doubly CI among both p and n, resulting in the highest possible exact-zero-value percentage in the precision matrix, alongside its lowest possible generation order. The thesis also explores the possibility of the co-existence of geostatistical and MRF modelling approaches in one unified framework, imparting a potential solution to an open problem. The derived theories are illustrated with 1D and 2D spatial data.