2015/05/30 by Joseph Guinness, Guinness, Joseph, Ilse C. F. Ipsen +1
Economics, Econometrics and Finance · Environmental Science · Mathematics · #Air Quality and Health Impacts #Computation (stat.CO) #Economic and Environmental Valuation #FOS: Computer and information sciences #Soil Geostatistics and Mapping #Statistical Methods and Bayesian Inference
paper · pdf · doi:10.48550/arxiv.1506.00138
openalex publication_date 2015/05/30 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
Rue and Held (2005) proposed a method for efficiently computing the Gaussian\nlikelihood for stationary Markov random field models, when the data locations\nfall on a complete regular grid, and the model has no additive error term. The\ncalculations rely on the availability of the covariances. We prove a theorem\ngiving the rate of convergence of a spectral method of computing the\ncovariances, establishing that the error decays faster than any polynomial in\nthe size of the computing grid. We extend the exact likelihood calculations to\nthe case of non-rectangular domains and missing values on the interior of the\ngrid and to the case when an additive uncorrelated error term (nugget) is\npresent in the model. We also give an alternative formulation of the likelihood\nthat has a smaller memory burden, parts of which can be computed in parallel.\nWe show in simulations that using the exact likelihood can give far better\nparameter estimates than using standard Markov random field approximations.\nHaving access to the exact likelihood allows for model comparisons via\nlikelihood ratios on large datasets, so as an application of the methods, we\ncompare several state-of-the-art methods for large spatial datasets on an\naerosol optical thickness dataset. We find that simple block independent\nlikelihood and composite likelihood methods outperform stochastic partial\ndifferential equation approximations in terms of computation time and returning\nparameter estimates that nearly maximize the likelihood.\n