Strategies for Fitting Large, Geostatistical Data in MCMC Simulation

Abstract

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.

Keywords:

• Band system solver
• Bayesian model
• Generalized linear mixed model
• Geostatistics
• Gibbs–Poole–Stockmeyer algorithm
• Incomplete factori-zation
• Markov chain Monte Carlo
• Quotient-minimum-degree algorithm
• Sweep operator

Mathematics Subject Classification:

• 62F15

Acknowledgment

This work was supported by the Swiss National Science Foundation Grant No. 3200–057165.99.