Dynamically updated spatially varying parameterizations of hierarchical Bayesian models for spatial data

Abstract

Fitting hierarchical Bayesian models to spatially correlated data sets using Markov chain Monte Carlo (MCMC) techniques is computationally expensive. Complicated covariance structures of the underlying spatial processes, together with high dimensional parameter space, mean that the number of calculations required grows cubically with the number of spatial locations at each MCMC iteration. This necessitates the need for efficient model parameterisations that hasten the convergence and improve the mixing of the associated algorithms. We consider partially centred parameterisations (PCPs) which lie on a continuum between what are known as the centred (CP) and noncentered parameterisations (NCP). By introducing a weight matrix we remove the conditional posterior correlation between the fixed and the random effects, and hence construct a PCP which achieves immediate convergence for a three stage model, based on multiple Gaussian processes with known covariance parameters. When the covariance parameters are unknown we dynamically update the parameterisation within the sampler. The PCP outperforms both the CP and the NCP and leads to a fully automated algorithm which has been demonstrated in two simulation examples. The effectiveness of the spatially varying PCP is illustrated with a practical data set of nitrogen dioxide concentration levels. Supplemental materials consisting of appendices, data sets and computer code to reproduce the results are available online.

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ORCID for Sujit Sahu: orcid.org/0000-0003-2315-3598

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