Sparse Log Gaussian Processes via MCMC for Spatial Epidemiology

In this work a fully independent training conditional (FITC) sparse approximation is used to speed up GP computations in the study of the spatial variations in relative mortality risk in a point referenced health-care data. The sampling of the latent values is sped up with transformations taking into account the approximate conditional posterior precision. Log Gaussian processes (LGP) are an attractive way to construct intensity surfaces for the purposes of spatial epidemiology. The spatial correlations between areas are included in an explicit and a natural way into the model via a covariance function. The drawback of GP is the computational burden of the required covariance matrix inversion. The computation time becomes prohibitive as the data amount increases up to around a few thousand of cases, limiting the study either to very small areas or a sparsely populated grid. To overcome the computational limitations a number of sparse approximations for GP have been suggested in the literature. Here a fully independent training conditional (FITC) sparse approximation is used to speed up GP computations. To set a golden standard for the uncertainty estimates, we integrate over both the hyperparameters and the latent values using Markov chain Monte Carlo methods (MCMC). The sampling of the latent values is sped up with transformations taking into account the approximate conditional posterior precision.