Statistical Methods & Applications

, Volume 25, Issue 1, pp 21–37 | Cite as

Covariance tapering for multivariate Gaussian random fields estimation

  • M. Bevilacqua
  • A. Fassò
  • C. Gaetan
  • E. Porcu
  • D. Velandia
Original Paper


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.


Cross Covariance estimation Large datasets Multivariate compactly supported correlation function Multivariate Gaussian process 

Supplementary material

10260_2015_338_MOESM1_ESM.pdf (217 kb)
Supplementary material 1 (pdf 217 KB)


  1. 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 97:15–30MathSciNetGoogle Scholar
  2. 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 21:75–91CrossRefMathSciNetGoogle Scholar
  3. Askey R (1973) Radial characteristic functions. Technical report, Research Center, University of WisconsinGoogle Scholar
  4. Bevilacqua M, Gaetan C (2015) Comparing composite likelihood methods based on pairs for spatial gaussian random fields. Stat Comput 25:877–892CrossRefMathSciNetGoogle Scholar
  5. 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 107:268–280CrossRefMathSciNetMATHGoogle Scholar
  6. Bevilacqua M, Hering A, Porcu E (2015) On the flexibility of multivariate covariance models: comment on the paper by Genton and Kleiber. Stat Sci 30:167–169CrossRefMathSciNetGoogle Scholar
  7. Daley D, Porcu E, Bevilacqua M (2015) Classes of compactly supported covariance functions for multivariate random fields. Stoch Environ Res Risk Assess 29:1249–1263CrossRefGoogle Scholar
  8. Du J, Zhang H, Mandrekar VS (2009) Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann Stat 37:3330–3361CrossRefMathSciNetMATHGoogle Scholar
  9. 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 23:295–315CrossRefMathSciNetGoogle Scholar
  10. Fontanella L, Ippoliti L (2003) Dynamic models for space-time prediction via karhunen-loeve expansion. Stat Methods Appl 12:61–78CrossRefMathSciNetMATHGoogle Scholar
  11. Furrer R, Sain SR (2010) spam: a sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random fields. J Stat Softw 36:1–25CrossRefGoogle Scholar
  12. Furrer R, Genton MG, Nychka D (2013) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15:502–523CrossRefMathSciNetGoogle Scholar
  13. Furrer R, Bachoc F, Du J (2015) Asymptotic properties of multivariate tapering for estimation and prediction. ArXiv e-prints arXiv:1506.01833
  14. Genton M, Kleiber W (2015) Cross-covariance functions for multivariate geostatistics. Stat Sci 30:147–163CrossRefMathSciNetGoogle Scholar
  15. Gneiting T (2002) Compactly supported correlation functions. J Multivar Anal 83:493–508CrossRefMathSciNetMATHGoogle Scholar
  16. Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177CrossRefMathSciNetGoogle Scholar
  17. Horn RA, Johnson CR (1991) Top matrix anal. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103:1545–1555CrossRefMathSciNetMATHGoogle Scholar
  19. Matheron G (1962) Traité de géostatistique appliquée, Tome 1. Mémoires du BRGM, n. 14, Technip, ParisGoogle Scholar
  20. Padoan S, Bevilacqua M (2015) Analysis of random fields using CompRandFld. J Stat Softw 63:1–27CrossRefGoogle Scholar
  21. Porcu E, Daley D, Buhmann M, Bevilacqua M (2013) Radial basis functions with compact support for multivariate geostatistics. Stoch Environ Res Risk Assess 27:909–922CrossRefGoogle Scholar
  22. Shaby B, Ruppert D (2012) Tapered covariance: Bayesian estimation and asymptotics. J Comput Graph Stat 21:433–452CrossRefMathSciNetGoogle Scholar
  23. Stein M, Chi Z, Welty L (2004) Approximating likelihoods for large spatial data sets. J R Stat Soc B 66:275–296CrossRefMathSciNetMATHGoogle Scholar
  24. Stein M, Chen J, Anitescu M (2012) Difference filter preconditioning for large covariance matrices. SIAM J Matrix Anal Appl 33:52–72CrossRefMathSciNetMATHGoogle Scholar
  25. Stein M, Chen J, Anitescu M (2013) Stochastic approximation of score functions for gaussian processes. Ann Appl Stat 7:1162–1191CrossRefMathSciNetMATHGoogle Scholar
  26. Vecchia A (1988) Estimation and model identification for continuous spatial processes. J R Stat Soc B 50:297–312MathSciNetGoogle Scholar
  27. Vetter P, Schmid W, Schwarze R (2015) Spatio-temporal statistical analysis of the carbon budget of the terrestrial ecosystem. Stat Methods ApplGoogle Scholar
  28. Wackernagel H (2003) Multivariate geostatistics: an introduction with applications, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  29. Zastavnyi V, Trigub R (2002) Positive definite splines of special form. Sbornik Math 193:1771–1800CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • M. Bevilacqua
    • 1
  • A. Fassò
    • 2
  • C. Gaetan
    • 3
  • E. Porcu
    • 4
  • D. Velandia
    • 1
  1. 1.Instituto de EstadísticaUniversidad de ValparaísoValparaisoChile
  2. 2.Dipartimento di IngegneriaUniversitá degli Studi di BergamoBergamoItaly
  3. 3.DAIS, Universitá Cá Foscari-VeneziaVeniceItaly
  4. 4.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile

Personalised recommendations