Matrix-Free Conditional Simulations of Gaussian Markov Random Fields

• Activity Number: 344
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistics and the Environment
• Author(s): Somak Dutta*+ and Debashis Mondal
• Companies: University of Chicago and University of Chicago
• Keywords: Discrete Cosine transformation ; Intrinsic autoregression ; Krylov subspace ; Lanczos algorithm ; Marginal likelihood ; Precision agriculture

Abstract

We develop a matrix-free method for conditional simulations for hidden Gaussian Markov random field models. For regular arrays, we exploit the analytic structure of the precision matrix using the two-dimensional discrete cosine transformation and employ the Lanczos algorithm to solve system of linear equations. As a key ingredient, we use an incomplete Cholesky factor of the sparse precision matrix as the preconditioner in the Lanczos algorithm and bring down the computation cost to O(rc log(rc)) for an r x c array. We provide further use of the conditional simulation to 1) quantify prediction uncertainty of the latent spatial field, 2) compute the marginal likelihood by Monte Carlo integrating over missing observations and 3) sample from the posterior distribution of the spatial field and other parameters under a Bayesian regime. We then demonstrate our method with a real data from an agricultural precision trial on a 480 x 1075 regular array. Finally, we discuss immediate extensions of our method for some other Gaussian random fields.