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-MRF model class, consisting of a mixed spatial graphical model framework and cross-MRF theory, to collectively address these challenges in one unified framework across two modelling stages. The first stage exploits scientifically informed conditional independence (CI) among p component fields and allows for a step-wise parallel generation of joint covariance and precision matrix, enabling a simultaneous accommodation of asymmetric cross-covariance in joint covariance matrix and sparsity in joint precision matrix. The second stage extends the first-stage CI to doubly CI among both p and n and unearths the cross-MRF via an extended Hammersley-Clifford theorem for multivariate spatial stochastic processes. This results in the sparsest possible representation of the joint precision matrix and ensures its lowest generation complexity. We demonstrate with 1D simulated comparative studies and 2D real-world data.

翻译：分析高维大规模空间随机过程时，其分量数量（p）与空间位置数量（n）均可很大，面临的关键挑战包括：实现联合精度矩阵的最大稀疏性、确保其生成的计算效率、适应联合协方差矩阵中的非对称交叉协方差，以及提供科学可解释性。我们提出了一类交叉马尔可夫随机场模型，它包含一个混合空间图模型框架和交叉马尔可夫随机场理论，旨在通过两个建模阶段在一个统一框架内共同应对这些挑战。第一阶段利用 p 个分量场之间基于科学知识的条件独立性，允许以逐步并行方式生成联合协方差矩阵和精度矩阵，从而能够同时适应联合协方差矩阵中的非对称交叉协方差以及联合精度矩阵的稀疏性。第二阶段将第一阶段的条件独立性扩展至 p 和 n 的双重条件独立性，并通过针对多元空间随机过程的扩展 Hammersley-Clifford 定理揭示了交叉马尔可夫随机场。这实现了联合精度矩阵最稀疏的可能表示，并确保了其最低的生成复杂度。我们通过一维模拟比较研究和二维真实世界数据进行了验证。