Abstract
Markov chain Monte Carlo (MCMC) algorithms for Bayesian computation for Gaussian process-based models under default parameterisations are slow to converge due to the presence of spatial- and other-induced dependence structures. The main focus of this paper is to study the effect of the assumed spatial correlation structure on the convergence properties of the Gibbs sampler under the default non-centred parameterisation and a rival centred parameterisation (CP), for the mean structure of a general multi-process Gaussian spatial model. Our investigation finds answers to many pertinent, but as yet unanswered, questions on the choice between the two. Assuming the covariance parameters to be known, we compare the exact rates of convergence of the two by varying the strength of the spatial correlation, the level of covariance tapering, the scale of the spatially varying covariates, the number of data points, the number and the structure of block updating of the spatial effects and the amount of smoothness assumed in a Matérn covariance function. We also study the effects of introducing differing levels of geometric anisotropy in the spatial model. The case of unknown variance parameters is investigated using well-known MCMC convergence diagnostics. A simulation study and a real-data example on modelling air pollution levels in London are used for illustrations. A generic pattern emerges that the CP is preferable in the presence of more spatial correlation or more information obtained through, for example, additional data points or by increased covariate variability.
Highlights
Correlated data are prevalent in many of the physical, biological and environmental sciences
We have compared the efficiencies of the centred parameterisation (CP) and the non-centred parameterisation (NCP) of spatial models
We find that in addition to the ratio of the variance parameters, the correlation structure between the random effects plays a key role in determining the rate of convergence
Summary
Correlated data are prevalent in many of the physical, biological and environmental sciences. We cast the general spatial model with multiple spatially varying covariates as a three stage normal linear hierarchical model This model formulation allows us to compute the exact rates of convergence for both CP and NCP for known prior covariance matrices by following (Roberts and Sahu 1997). A direct theoretical comparison between the exact convergence rates of the centring parameterisations for Gaussian process (GP)-based models and the DA algorithms is desirable, as has been done by Sahu and Roberts (1999) for the EM algorithm and the Gibbs sampler. It studies the effect of tapering and the scale of the spatially varying covariates on the rates of convergence. Appendices A and B, respectively, contain the technical details for calculating the rates of convergence and the full conditional distributions needed for Gibbs sampling
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