Fully Bayesian inference for spatiotemporal data with the multi-resolution approximation.

Large spatiotemporal datasets are a challenge for conventional Bayesian models because of the cubic computational complexity of the algorithms for obtaining the Cholesky decomposition of the covariance matrix in the multivariate normal density. Moreover, standard numerical algorithms for posterior estimation, such as Markov Chain Monte Carlo (MCMC), are intractable in this context, as they require thousands, if not millions, of costly likelihood evaluations. To overcome those limitations, we propose IS-MRA (Importance sampling - Multi-Resolution Approximation), which takes advantage of the sparse inverse covariance structure produced by the Multi-Resolution Approximation (MRA) approach. IS-MRA is fully Bayesian and facilitates the approximation of the hyperparameter marginal posterior distributions. We apply IS-MRA to large MODIS Level 3 Land Surface Temperature (LST) datasets, sampled between May 18 and May 31, 2012 in the western part of the state of Maharashtra, India. We find that IS-MRA can produce realistic prediction surfaces over regions where concentrated missingness, caused by sizable cloud cover, is observed. Through a validation analysis and simulation study, we also find that predictions tend to be very accurate.