Covariance Tapering for Likelihood Based Estimation in Large Spatial Datasets
Cari Kaufman

Monday, November 6, 2006
4:10 p.m.-5:00 p.m.
203 Engineering


Likelihood-based methods such as maximum likelihood, REML, and
Bayesian methods are attractive approaches to estimating covariance
parameters in spatial models based on Gaussian processes. Finding such
estimates can be computationally infeasible for large datasets,
however, requiring O(n 3 ) calculations for each evaluation of the
likelihood based on n observations. I propose the method of covariance
tapering to approximate the likelihood in this setting. In this
approach, covariance matrices are "tapered," or multiplied
element-wise by a sparse correlation matrix. This produces matrices
which can be be manipulated using more efficient sparse matrix
algorithms. I present two approximations to the Gaussian likelihood
using tapering. Focusing on the particular case of the Matern class of
covariance functions, I give conditions under which tapered and
untapered covariance functions produce equivalent (mutually absolutely
continuous) measures for Gaussian processes on bounded domains. This
allows me to evaluate the behavior of estimators maximizing the
approximations to the likelihood under a bounded domain asymptotic
framework. I present results from a simulation study showing agreement
between the asymptotic results and what we observe for moderate but
increasing sample sizes. Tapering methods can also be applied in
fitting hierarchical Bayesian models involving large multivariate
normal densities. Here, I discuss the particular application of making
inference about the climatological (long-run mean) temperature
difference between two sets of output from a computer model of global
climate, run under two different land use scenarios.



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