Tapered Covariance: Bayesian Estimation and Asymptotics

Benjamin Shaby, David Ruppert
2012 Journal of Computational And Graphical Statistics  
The method of maximum tapered likelihood has been proposed as a way quickly to estimate covariance parameters for stationary Gaussian random fields. We show that under a useful asymptotic regime, maximum tapered likelihood estimators are consistent and asymptotically normal for covariance models in common use. We then formalize the notion of tapered quasi-Bayesian estimators and show that they too are consistent and asymptotically normal. We also present asymptotic confidence intervals for both
more » ... types of estimators and show via simulation that they accurately reflect sampling variability, even at modest sample sizes. Proofs, an example, and detailed derivations are found in the supplemental materials, available online. Covariance tapering was introduced as a way to mitigate the computational burdens required for calculating statistically-relevant quantities involving large covariance matrices arising from irregularly-spaced spatial data. These computations typically require O(n 3 ) operations, where n is the number of spatial observations. The idea behind tapering is to introduce, in a principled way, many zeros into the covariance matrices, enabling the use of sparse matrix algorithms, which have computational complexities that are generally functions of the number of non-zero elements in the matrix. Tapering has been studied as a way to speed up computations required for optimal spatial prediction (Furrer et al. 2006; Furrer and Sain 2009) and for Kalman filter updates (Furrer and Bengtsson 2007). Kaufman (2006) and Kaufman et al. (2008) introduced the maximum tapered likelihood estimate as a way to use tapered covariance matrices to quickly estimate covariance function parameters. Du et al. (2009) and Zhang and Du (2008) further explicated the properties of these estimators. In addition, Kaufman (2006) discussed approximating Bayesian estimation using tapered likelihood functions. Here, we examine the behavior of maximum tapered likelihood estimators,
doi:10.1080/10618600.2012.680819 fatcat:wkuzolwptfbcziz6gnyimlglfa