Abstract
In recent literature there has been a growing interest in the construction of covariance models for multivariate Gaussian random fields. However, effective estimation methods for these models are somehow unexplored. The maximum likelihood method has attractive features, but when we deal with large data sets this solution becomes impractical, so computationally efficient solutions have to be devised. In this paper we explore the use of the covariance tapering method for the estimation of multivariate covariance models. In particular, through a simulation study, we compare the use of simple separable tapers with more flexible multivariate tapers recently proposed in the literature and we discuss the asymptotic properties of the method under increasing domain asymptotics.
Similar content being viewed by others
References
Apanasovich T, Genton M, Sun Y (2012) A valid Matérn class of cross-covariance functions for multivariate random fields with any number of components. J Am Stat Assoc 97:15–30
Arima S, Cretarola L, Lasinio GJ, Pollice A (2012) Bayesian univariate space-time hierarchical model for mapping pollutant concentrations in the municipal area of taranto. Stat Methods Appl 21:75–91
Askey R (1973) Radial characteristic functions. Technical report, Research Center, University of Wisconsin
Bevilacqua M, Gaetan C (2015) Comparing composite likelihood methods based on pairs for spatial gaussian random fields. Stat Comput 25:877–892
Bevilacqua M, Gaetan C, Mateu J, Porcu E (2012) Estimating space and space-time covariance functions for large data sets: a weighted composite likelihood approach. J Am Stat Assoc 107:268–280
Bevilacqua M, Hering A, Porcu E (2015) On the flexibility of multivariate covariance models: comment on the paper by Genton and Kleiber. Stat Sci 30:167–169
Daley D, Porcu E, Bevilacqua M (2015) Classes of compactly supported covariance functions for multivariate random fields. Stoch Environ Res Risk Assess 29:1249–1263
Du J, Zhang H, Mandrekar VS (2009) Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann Stat 37:3330–3361
Eidsvik J, Shaby BA, Reich BJ, Wheeler M, Niemi J (2014) Estimation and prediction in spatial models with block composite likelihoods. J Comput Graph Stat 23:295–315
Fontanella L, Ippoliti L (2003) Dynamic models for space-time prediction via karhunen-loeve expansion. Stat Methods Appl 12:61–78
Furrer R, Sain SR (2010) spam: a sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random fields. J Stat Softw 36:1–25
Furrer R, Genton MG, Nychka D (2013) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15:502–523
Furrer R, Bachoc F, Du J (2015) Asymptotic properties of multivariate tapering for estimation and prediction. ArXiv e-prints arXiv:1506.01833
Genton M, Kleiber W (2015) Cross-covariance functions for multivariate geostatistics. Stat Sci 30:147–163
Gneiting T (2002) Compactly supported correlation functions. J Multivar Anal 83:493–508
Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177
Horn RA, Johnson CR (1991) Top matrix anal. Cambridge University Press, Cambridge
Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103:1545–1555
Matheron G (1962) Traité de géostatistique appliquée, Tome 1. Mémoires du BRGM, n. 14, Technip, Paris
Padoan S, Bevilacqua M (2015) Analysis of random fields using CompRandFld. J Stat Softw 63:1–27
Porcu E, Daley D, Buhmann M, Bevilacqua M (2013) Radial basis functions with compact support for multivariate geostatistics. Stoch Environ Res Risk Assess 27:909–922
Shaby B, Ruppert D (2012) Tapered covariance: Bayesian estimation and asymptotics. J Comput Graph Stat 21:433–452
Stein M, Chi Z, Welty L (2004) Approximating likelihoods for large spatial data sets. J R Stat Soc B 66:275–296
Stein M, Chen J, Anitescu M (2012) Difference filter preconditioning for large covariance matrices. SIAM J Matrix Anal Appl 33:52–72
Stein M, Chen J, Anitescu M (2013) Stochastic approximation of score functions for gaussian processes. Ann Appl Stat 7:1162–1191
Vecchia A (1988) Estimation and model identification for continuous spatial processes. J R Stat Soc B 50:297–312
Vetter P, Schmid W, Schwarze R (2015) Spatio-temporal statistical analysis of the carbon budget of the terrestrial ecosystem. Stat Methods Appl
Wackernagel H (2003) Multivariate geostatistics: an introduction with applications, 3rd edn. Springer, New York
Zastavnyi V, Trigub R (2002) Positive definite splines of special form. Sbornik Math 193:1771–1800
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Bevilacqua, M., Fassò, A., Gaetan, C. et al. Covariance tapering for multivariate Gaussian random fields estimation. Stat Methods Appl 25, 21–37 (2016). https://doi.org/10.1007/s10260-015-0338-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-015-0338-3