Composite Likelihood Inference for Multivariate Gaussian Random Fields

Abstract

In the recent years, there has been a growing interest in proposing covariance models for multivariate Gaussian random fields. Some of these covariance models are very flexible and can capture both the marginal and the cross-spatial dependence of the components of the associated multivariate Gaussian random field. However, effective estimation methods for these models are somehow unexplored. Maximum likelihood is certainly a useful tool, but it is impractical in all the circumstances where the number of observations is very large. In this work, we consider two possible approaches based on composite likelihood for multivariate covariance model estimation. We illustrate, through simulation experiments, that our methods offer a good balance between statistical efficiency and computational complexity. Asymptotic properties of the proposed estimators are assessed under increasing domain asymptotics. Finally, we apply the method for the analysis of a bivariate dataset on chlorophyll concentration and sea surface temperature in the Chilean coast.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Apanasovich, T., Genton, M., and Sun, Y. (2012), “A valid Matérn class of cross-covariance functions for multivariate random fields with any number of components,” Journal of the American Statistical Association, 97, 15–30.

    MathSciNet  MATH  Google Scholar 

  2. Bevilacqua, M., Fassò, A., Gaetan, C., Porcu, E., and Velandia, D. (2016), “Covariance tapering for multivariate Gaussian random fields estimation,” Statistical Methods & Applications, 25(1), 21–37.

    MathSciNet  Article  MATH  Google Scholar 

  3. Bevilacqua, M., and Gaetan, C. (2015), “Comparing composite likelihood methods based on pairs for spatial Gaussian random fields,” Statistics and Computing, 25, 877–892.

    MathSciNet  Article  MATH  Google Scholar 

  4. Bevilacqua, M., Gaetan, C., Mateu, J., and Porcu, E. (2012), “Estimating space and space-time covariance functions for large data sets: a weighted composite likelihood approach,” Journal of the American Statistical Association, 107, 268–280.

    MathSciNet  Article  MATH  Google Scholar 

  5. Bevilacqua, M., Vallejos, R., and Velandia, D. (2015), “Assessing the significance of the correlation between the components of a bivariate Gaussian random field,” Environmetrics, 26, 545–556.

    MathSciNet  Article  Google Scholar 

  6. Boyce, D. G., Lewis, M. R., and Worm, B. (2010), “Global phytoplankton decline over the past century,” Nature. International weekly journal of science, 466. doi:10.1038/nature09268.

  7. Castruccio, S., Huser, R., and Genton, M. G. (2016), “High-order composite likelihood inference for max-stable distributions and processes,” Journal of Computational and Graphical Statistics. To appear.

  8. Daley, D., Porcu, E., and Bevilacqua, M. (2015), “Classes of compactly supported covariance functions for multi- variate random fields,” Stoch Environ Res Risk Assess, 29, 1249–1263.

    Article  Google Scholar 

  9. Davis, R., and Yau, C.-Y. (2011), “Comments on pairwise likelihood in time series models,” Statistica Sinica, 21, 255–277.

    MathSciNet  MATH  Google Scholar 

  10. Doney, S. C., Ruckelshaus, M., Duffy, J. E., Barry, J. P., Chan, F., English, C. A., Galindo, H. M., Grebmeier, J. M., Hollowed, A. B., Knowlton, N., Polovina, J., Rabalais, N. N., Sydeman, W. J., and Talley, L. D. (2012), “Annual Review of Marine Science,” Nature. International weekly journal of science, 4, 11–37.

    Google Scholar 

  11. Eidsvik, J., Shaby, B., Reich, B., Wheeler, M., and Niemi, J. (2014), “Estimation and prediction in spatial models with block composite likelihoods,” Journal of Computational and Graphical Statistics, 29, 295–315.

    MathSciNet  Article  Google Scholar 

  12. Furrer, R., Bachoc, F., and Du, J. (2016), “Asymptotic properties of multivariate tapering for estimation and prediction,” Journal of Multivariate Analysis, In press.

  13. Furrer, R., Genton, M. G., and Nychka, D. (2006), “Covariance tapering for interpolation of large spatial datasets,” Journal of Computational and Graphical Statistics, 15, 502–523.

    MathSciNet  Article  Google Scholar 

  14. Genton, M. G., Padoan, S., and Sang, H. (2015), “Multivariate max-stable spatial processes,” Biometrika, 102, 215 –230.

    MathSciNet  Article  MATH  Google Scholar 

  15. Genton, M., and Kleiber, W. (2015), “Cross-Covariance Functions for Multivariate Geostatistics,” Statistical Science, in press.

  16. Gneiting, T. (2002), “Compactly supported correlation functions,” Journal of Multivariate Analysis, 83, 493–508.

    MathSciNet  Article  MATH  Google Scholar 

  17. Gneiting, T., Genton, M. G., and Guttorp, P. (2007), “Geostatistical space-time models, stationarity, separability and full symmetry,” in Statistical Methods for Spatio-Temporal Systems, eds. B. Finkenstadt, L. Held, and V. Isham, Boca Raton: FL: Chapman & Hall/CRC, pp. 151–175.

    Google Scholar 

  18. Gneiting, T., Kleiber, W., and Schlather, M. (2010), “Matérn Cross-Covariance Functions for Multivariate Random Fields,” Journal of the American Statistical Association, 105, 1167–1177.

    MathSciNet  Article  MATH  Google Scholar 

  19. Goulard, M., and Voltz, M. (1992), “Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix,” Mathematical Geology, 24, 269–286.

    Article  Google Scholar 

  20. Heagerty, P., and Lumley, T. (2000), “Window subsampling of estimating functions with application to regression models,” Journal of the American Statistical Association, 95, 197–211.

    MathSciNet  Article  MATH  Google Scholar 

  21. Joe, H., and Lee, Y. (2009), “On weighting of bivariate margins in pairwise likelihood,” Journal of Multivariate Analysis, 100, 670–685.

    MathSciNet  Article  MATH  Google Scholar 

  22. Kaufman, C. G., Schervish, M. J., and Nychka, D. W. (2008), “Covariance tapering for likelihood-based estimation in large spatial data sets,” Journal of the American Statistical Association, 103, 1545–1555.

    MathSciNet  Article  MATH  Google Scholar 

  23. Lee, A., Yau, C., Giles, M., Doucet, A., and Holmes, C. (2010), “On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods,” Journal of Computational and Graphical Statistics, 19, 769 –789.

    Article  Google Scholar 

  24. Lee, Y., and Lahiri, S. (2002), “Least squares variogram fitting by spatial subsampling,” Journal of the Royal Statistical Society B, 64, 837–854.

    MathSciNet  Article  MATH  Google Scholar 

  25. Li, B., and Zhang, H. (2011), “An approach to modeling asymmetric multivariate spatial covariance structures,” Journal of Multivariate Analysis, 102, 1445–1453.

    MathSciNet  Article  MATH  Google Scholar 

  26. Lindsay, B. (1988), “Composite likelihood methods,” Contemporary Mathematics, 80, 221–239.

    MathSciNet  Article  MATH  Google Scholar 

  27. Padoan, S. A., and Bevilacqua, M. (2015), “Analysis of Random Fields Using CompRandFld,” Journal of Statistical Software, 63, 1–27.

    Article  Google Scholar 

  28. Pelletier, B., Dutilleul, P., Larocque, G., and Fyles, J. (2004), “Fitting the linear model of coregionalization by generalized least squares,” Mathematical Geology, 36(3), 323–343.

    Article  MATH  Google Scholar 

  29. Porcu, E., Daley, D., Buhmann, M., and Bevilacqua, M. (2013), “Radial basis functions with compact support for multivariate geostatistics,” Stochastic Environmental Research and Risk Assessment, 27, 909–922.

    Article  Google Scholar 

  30. Shaby, B., and Ruppert, D. (2012), “Tapered covariance: Bayesian estimation and asymptotics,” Journal of Computational and Graphical Statistics, 21, 433–452.

    MathSciNet  Article  Google Scholar 

  31. Stein, M. (2005), “Space-time covariance functions,” Journal of the American Statistical Association, 100, 310–321.

    MathSciNet  Article  MATH  Google Scholar 

  32. Stein, M., Chi, Z., and Welty, L. (2004), “Approximating likelihoods for large spatial data sets,” Journal of the Royal Statistical Society B, 66, 275–296.

    MathSciNet  Article  MATH  Google Scholar 

  33. Suchard, M., Wang, Q., anf J. Frelinger, C. C., Cron, A., and West, M. (2010), “Understanding GPU programming for statistical computation: studies in massively parallel massive mixtures,” Journal of Computational and Graphical Statistics, 19, 419 –438.

    MathSciNet  Article  Google Scholar 

  34. Varin, C., Reid, N., and Firth, D. (2011), “An overview of composite likelihood methods,” Statistica Sinica, 21, 5–42.

    MathSciNet  MATH  Google Scholar 

  35. Varin, C., and Vidoni, P. (2005), “A note on composite likelihood inference and model selection,” Biometrika, 52, 519–528.

    MathSciNet  Article  MATH  Google Scholar 

  36. Wackernagel, H. (2003), Multivariate Geostatistics: An Introduction with Applications, 3rd edn, New York: Springer.

    Book  MATH  Google Scholar 

  37. Wood, S. (2006), Generalized Additive Models: An Introduction with R, : Chapman and Hall CRC.

    MATH  Google Scholar 

  38. Zhang, H. (2007), “Maximum-likelihood estimation for multivariate spatial linear coregionalization models,” Environmetrics, 18, 125–139.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The research work conducted by Moreno Bevilacqua was supported in part by FONDECYT Grant 11121408, Chile. Emilio Porcu has been supported by FONDECYT Grant 1130647.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Moreno Bevilacqua.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 235 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bevilacqua, M., Alegria, A., Velandia, D. et al. Composite Likelihood Inference for Multivariate Gaussian Random Fields. JABES 21, 448–469 (2016). https://doi.org/10.1007/s13253-016-0256-3

Download citation

Keywords

  • Cross-covariance
  • Large datasets
  • Geostatistics