Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xiaoqing Chen, Peter Diggle, James V. Zidek, Gavin Shaddick

TL;DR

This work introduces a cross-Markov Random Field (cross-MRF) framework to tackle highly multivariate large-scale spatial processes by jointly learning a sparse precision matrix and an asymmetric cross-covariance structure. The approach pairs a first-stage CI-based mixed spatial graphical model with a second-stage cross-MRF that enforces doubly CI across both components and locations, achieving maximal sparsity in $\Sigma^{-1}_{np\times np}$ and reducing generation complexity to $\mathcal{O}(pn^2)$. The methodology is supported by theoretical extensions of the Hammersley-Clifford theorem to multivariate spatial processes and concrete linking strategies via moralisation and CAR-based univariate inverses. Empirical results on 1D simulations and 2D CAMS data demonstrate accurate capture of asymmetric cross-covariance, scalable computation, and interpretable neighborhood structures. The framework offers practical benefits for large-scale environmental and socio-economic spatio-temporal analyses, with clear avenues for distributed inference and spatio-temporal generalisation.

Abstract

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.

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

TL;DR

This work introduces a cross-Markov Random Field (cross-MRF) framework to tackle highly multivariate large-scale spatial processes by jointly learning a sparse precision matrix and an asymmetric cross-covariance structure. The approach pairs a first-stage CI-based mixed spatial graphical model with a second-stage cross-MRF that enforces doubly CI across both components and locations, achieving maximal sparsity in and reducing generation complexity to . The methodology is supported by theoretical extensions of the Hammersley-Clifford theorem to multivariate spatial processes and concrete linking strategies via moralisation and CAR-based univariate inverses. Empirical results on 1D simulations and 2D CAMS data demonstrate accurate capture of asymmetric cross-covariance, scalable computation, and interpretable neighborhood structures. The framework offers practical benefits for large-scale environmental and socio-economic spatio-temporal analyses, with clear avenues for distributed inference and spatio-temporal generalisation.

Abstract

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.
Paper Structure (38 sections, 8 theorems, 49 equations, 12 figures, 9 tables, 2 algorithms)

This paper contains 38 sections, 8 theorems, 49 equations, 12 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions of Mixed Probabilistic Spatial Graphs
  3. Mixed Spatial Graphical Model Framework
  4. The first-stage model: CI among $p$ component fields
  5. The algorithm
  6. 1D Simulation: Ten component Fields
  7. Beyond CI among p-component Fields: Cross-MRF
  8. H-C theorem for multivariate spatial stochastic processes
  9. Conditions
  10. Cross-MRF
  11. Linking the Cross-MRF and Mixed Spatial Graphical Model Framework
  12. Realising symmetry between component fields
  13. Realising $s_j \in \mathcal{N}(s_i)$
  14. 1D Comparative Study
  15. 2D Illustration: CAMS Reanalysis Data
  16. ...and 23 more sections

Key Result

Theorem 1

For any customised structure among component fields in the mixed spatial graph, the joint covariance matrix $\hbox{$\Sigma$}_{np \times np}$ for any $p$, $p \triangleq j+1$, can be obtained using the updating formula: $\hbox{$\Sigma$}_{np \times np} \triangleq \hbox{$\Sigma$}_{(j+1)n \times (j+1)n} where $\bm{B}_{(j+1)k^c}(\cdot, \cdot) \neq \bm{0}$ for $k^c \in Pa(j+1)$. $k = \{1, \cdots, j \}$,

Figures (12)

  • Figure 1: (a) and (b) show the auto-correlation of DU and SU residuals after removing a quadratic trend surface; both are symmetric. (c) shows the cross-correlation between DU and SU residuals, which is asymmetric. Each plot spans $-15$ to $+15$ degrees in both coordinate directions.
  • Figure 2: Schematic representations of the row-wise conditional and column-wise conditional method. Row-wise conditional only regresses on values at neighbourhood locations but across all components, while column-wise conditional regresses on all the previous components across all spatial locations.
  • Figure 3: Illustration of a mixed probabilistic spatial graph. Across different component fields $\bm{Y}_k(\cdot)$ and $\bm{Y}_l(\cdot)$, nodes are connected using directed edges. Within a component field $\bm{Y}_k(\cdot)$, nodes are connected by undirected edges. Each node is a random quantity. Altogether, they consist of a mixed probabilistic spatial graph.
  • Figure 4: A hypothetically organised acyclic graphical structure for ten fields. Each parallelogram represents a component field.
  • Figure 5: 1D simulation of $\Sigma$ and $\Sigma^{-1}$ for ten component fields ($p = 10$). $b(\cdot, \cdot)$ function is modelled using a modified tri-wave function. The left figure is the joint covariance matrix $\Sigma$ and the right one is the joint precision matrix $\Sigma^{-1}$, on a log scale. Both $\textbf{B}_{rt}$ are under a combination of spectral normalisation (SpN) and regularisation (Reg) transformation. Parameter A = 0.1, $\Delta$ = 0.5. The largest possible threshold for $\Sigma^{-1}$ is $1e^{-3}$, the smallest possible regularisation number is $1e^{-9}$.
  • ...and 7 more figures

Theorems & Definitions (31)

  • Definition 1: Nodes of a probabilistic spatial graph
  • Definition 2: component field
  • Definition 3: Probabilistic spatial graph -- undirected
  • Definition 4: Component field spatial graph -- directed
  • Definition 5: Mixed probabilistic spatial graph
  • Theorem 1: Graph-structure-guided Updating Formula for $\hbox{$\Sigma$}_{np \times np}$
  • proof
  • Proposition 1: Positive Definite Condition for $\hbox{$\Sigma$}_{np \times np}$
  • proof
  • Theorem 2: Step-wise Construction of $\hbox{$\Sigma$}^{-1}_{np \times np}$
  • ...and 21 more