Valid parameter space of a bivariate Gaussian Markov random field with a generalized block-Toeplitz precision matrix

TL;DR

This paper characterizes the valid parameter space ensuring positive-definiteness of a bivariate Gaussian Markov random field with a generalized block-Toeplitz precision matrix, extending classical results to more complex spatial models.

Contribution

It provides asymptotic closed-form expressions for the valid parameter space of bivariate GMRFs with generalized block-Toeplitz precision matrices, extending classical convergence results.

Findings

Derived asymptotic expressions for parameter space

Developed a methodology for generalized block-Toeplitz structures

Quantified convergence rate through numerical experiments

Abstract

Gaussian Markov random fields (GMRFs) are extensively used in statistics to model area-based data and usually depend on several parameters in order to capture complex spatial correlations. In this context, it is important to determine the valid parameter space, namely the domain ensuring (semi) positive-definiteness of the precision matrix. Depending on the structure of the latter, this task can be challenging. While univari- ate GMRFs with block-Toeplitz precision are well studied in the literature, not much is analytically known about bivariate GMRFs. So far, only restrictive sufficient conditions and brute-force approaches were proposed, which are computationally expensive for the size of modern datasets. In this paper, we consider a bivariate GMRF, which is part of a hierarchical model used in spatial statistics to analyze data coming from projec- tions of regional climate change. By extending classical convergence results of univariate fields with toroidal boundary conditions to fields without boundary conditions, we pro- vide asymptotically closed-form expressions of the valid parameter space. We develop a general methodology that can be used to determine the valid parameter space of bivariate GMRFs whose precision matrix has a generalized block-Toeplitz structure and for which classical convergence results are not directly applicable. Finally, we quantify the rate of convergence of our approach through a numerical study in R.