Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets

By Cari Kaufman, Mark J. Schervish, D. Nychka. 2008

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Maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on Gaussian processes. But calculating the likelihood can be computationally infeasible for large data sets, requiring O(n3) calculations for a data set with n observations. This article proposes the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrixes are “tapered,” or multiplied element wise by a sparse correlation matrix. The resulting matrixes can then be manipulated using efficient sparse matrix algorithms. We propose two approximations to the Gaussian likelihood using tapering. One of these approximations simply replaces the model covariance with a tapered version, whereas the other is motivated by the theory of unbiased estimating equations. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which estimators maximizing the tapering approximations are, like the maximum likelihood estimator, strongly consistent. Moreover, we show in a simulation study that the tapering estimators can have sampling densities quite similar to that of the maximum likelihood estimator, even when the degree of tapering is severe. We illustrate the accuracy and computational gains of the tapering methods in an analysis of yearly total precipitation anomalies at weather stations in the United States.

Subfields: Economics and Econometrics, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8651ebf47bc5d3a2ab28c6ee928bdbf729a0ebf1

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Citations in the corpus

Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets (2014, methodology): “Methods exploiting sparse covariance structures include covariance tapering methods (see, e.g., Furrer et al. 2006; Kaufman et al. 2008; Du et al. 2009) or likelihood approximations using Markov random fields (e.”
Likelihood approximation with hierarchical matrices for large spatial datasets (2017, methodology): “…methods: for example, low-rank tensor methods (Litvinenko et al., 2018; Nowak and Litvinenko, 2013), covariance tapering (Furrer et al., 2006; Kaufman et al., 2008; Sang and Huang, 2012), likelihood approximations in both the spatial (Stein et al., 2004, 2013) and spectral (Fuentes, 2007)…”
Tapered Covariance: Bayesian Estimation and Asymptotics (2012, methodology): “Here, like Kaufman (2006), Kaufman, Schervish, and Nychka (2008), Zhang and Du (2008), and Du, Zhang, and Mandrekar (2009), we study the use of tapering for parameter estimation.”
Classes of compactly supported covariance functions for multivariate random fields (2015, methodology): “With these two ingredients, it will be now possible to extend the works of Du et al. (2009), Kaufman et al. (2008), and Furrer et al. (2006) to the multivariate case.”
The role of the range parameter for estimation and prediction in geostatistics (2011, methodology): “A similar method of proof can be used to show consistency and asymptotic normality of the maximum tapered likelihood estimator proposed by Kaufman et al. (2008).”
Joint composite estimating functions in spatiotemporal models (2012, methodology): “(c) The pairwise CL approach does not require a distance metric accommodating both space and time, in contrast with the unified distance norm that is needed by the tapering approach (Kaufman et al., 2008).”
Statistical Properties of Covariance Tapers (2013, methodology): “Then the first approximation in Kaufman, Schervish, and Nychka (2008) to the log-likelihood is 1(θ) = −1 2 log |C̃(θ )| − 1 2 Z′C̃(θ)−1 Z. (8) Kaufman, Schervish, and Nychka (2008) showed that estimating θ by maximizing 1(θ ) can sometimes lead to estimates with substantial bias (Kaufman,…”
A full scale approximation of covariance functions for large spatial data sets (2012, methodology): “If the tapered part is void, we obtain the predictive process of Banerjee et al. (2008); if the set SÅ of locations is empty, then the predictive process part is void and we obtain the tapered version of the original covariance function as used in Furrer et al. (2006) and Kaufman et al. (2008).”
Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe (2011, methodology): “In that circumstance, we may resort to covariance tapering (Kaufman et al., 2008; Furrer & Sain, 2009) or other likelihood approximation methods such as Stein et al. (2004) and Fuentes (2007).”
Fixed-domain asymptotic properties of tapered maximum likelihood estimators (2009, methodology): “Kaufman, et al. (2008) showed that the parameter in the Mat¶ern covariance function,”
On the consistent separation of scale and variance for Gaussian random fields (2009, background): “d all the parameters are consistently estimable when d4. Our results conrm these ndings for and ˙with the non-periodic Matern when d>4. The case d= 4 is still open. Recent work by Kaufman et al.[8] and Du et al.[13] studies maximum likelihood estimates of ˙22 using a tapered Matern autocovariance”
Estimating Space and Space-Time Covariance Functions for Large Data Sets: A Weighted Composite Likelihood Approach (2012, methodology): “In this sense, the method is different from other estimation methods of covariance functions (Stein, Chi, and Welty 2004; Kaufman et al. 2008).”
A General Framework for Vecchia Approximations of Gaussian Processes (2017, background): “These include approaches leading to sparse co-variance matrices (Furrer et al., 2006; Kaufman et al., 2008; Du et al., 2009), sparse inverse 2 A general Vecchia approach 2.1 Noisy observations of a Gaussian process Let { y ( s ) : s ∈ D} , or y ( · ), be a process of interest on a continuous (i.e.,…”