Approximating Likelihoods for Large Spatial Data Sets

Abstract

Likelihood methods are often difficult to use with large, irregularly sited spatial data sets, owing to the computational burden. Even for Gaussian models, exact calculations of the likelihood for "n" observations require "O"("n"-super-3) operations. Since any joint density can be written as a product of conditional densities based on some ordering of the observations, one way to lessen the computations is to condition on only some of the 'past' observations when computing the conditional densities. We show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation. Previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating. Through theoretical, numerical and practical examples, we show that there can often be considerable benefit in conditioning on some distant observations as well. Copyright 2004 Royal Statistical Society.

Interpolation of spatial data – A stochastic or a deterministic problem?

“…When all sampling locations are on a (near-)regular lattice, spectral methods to approximate the likelihood can be used and allow to reduce the computational cost to an order of O(n log(n)) [31,13,28]. These techniques cannot be applied to scattered data, but other approaches to approximating likelihoods [79,78,5,46], covariance tapering [29], or simplified Gaussian models of low rank [3,12,20] have been proposed and have been shown to be quite effective in reducing the computational effort to an order that allows the application of REML in most practical situations.…”

An alternative procedure for selecting a good value for the parameter c in RBF-interpolation

“…The M data points chosen from X n−1 are typically those closest to x n . If the optimal scale is expected to be large, however, the approximation can be improved considerably if some points further apart are also included (see [17] for a detailed discussion).…”

Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

“…CL methods are an attractive option when the full likelihood is difficult to write and/or when the data sets are large. This approach has been used in several spatial and space-time contexts, mainly in the Gaussian case (Vecchia, 1988;Curriero and Lele, 1999;Stein et al, 2004;Bevilacqua et al, 2012;Bevilacqua and Gaetan, 2015;Bevilacqua et al, 2016). Outside the Gaussian scenario, Heagerty and Lele (1998) propose CL inference for binary spatial data.…”