Dynamically Updated Spatially Varying Parameterizations of Hierarchical Bayesian Models for Spatial Data

ABSTRACT

Fitting hierarchical Bayesian models to spatially correlated datasets 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 parameterizations that hasten the convergence and improve the mixing of the associated algorithms. We consider partially centred parameterizations (PCPs) which lie on a continuum between what are known as the centered (CP) and noncentered parameterizations (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 parameterization 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 dataset of nitrogen dioxide concentration levels. Supplemental materials consisting of appendices, datasets, and computer code to reproduce the results are available online.

KEYWORDS:

• Gibbs sampling
• Parameterization
• Rate of convergence
• Spatially varying coefficient model

Supplementary Materials

	Appendices: A.1 to A.6 containing proofs and further details as referenced in the main article.
	C code: All of the C code used to generate the MCMC output which is analyzed to produce the results given in this article are contained within the folder called c_files. Full details are provided within the README.txt file contained within the folder.
	Data: All data files can be found in the folder labeled data_files. A full description is given in the README.txt file contained therein.
	R code: The R code used to analyze the MCMC output produced by the C code can be found in the folder labeled R_scripts. A full description of the files is given in README.txt.
	Figures: The figures produced by the R code included in this article are given in the folder labeled figures.

Notes

1 https://uk-air.defra.gov.uk/air-pollution/uk-eu-limits, last modified March 3, 2017