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.
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References
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.
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.
Bevilacqua, M., and Gaetan, C. (2015), “Comparing composite likelihood methods based on pairs for spatial Gaussian random fields,” Statistics and Computing, 25, 877–892.
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.
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.
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.
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.
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.
Davis, R., and Yau, C.-Y. (2011), “Comments on pairwise likelihood in time series models,” Statistica Sinica, 21, 255–277.
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.
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.
Furrer, R., Bachoc, F., and Du, J. (2016), “Asymptotic properties of multivariate tapering for estimation and prediction,” Journal of Multivariate Analysis, In press.
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.
Genton, M. G., Padoan, S., and Sang, H. (2015), “Multivariate max-stable spatial processes,” Biometrika, 102, 215 –230.
Genton, M., and Kleiber, W. (2015), “Cross-Covariance Functions for Multivariate Geostatistics,” Statistical Science, in press.
Gneiting, T. (2002), “Compactly supported correlation functions,” Journal of Multivariate Analysis, 83, 493–508.
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.
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.
Goulard, M., and Voltz, M. (1992), “Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix,” Mathematical Geology, 24, 269–286.
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.
Joe, H., and Lee, Y. (2009), “On weighting of bivariate margins in pairwise likelihood,” Journal of Multivariate Analysis, 100, 670–685.
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.
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.
Lee, Y., and Lahiri, S. (2002), “Least squares variogram fitting by spatial subsampling,” Journal of the Royal Statistical Society B, 64, 837–854.
Li, B., and Zhang, H. (2011), “An approach to modeling asymmetric multivariate spatial covariance structures,” Journal of Multivariate Analysis, 102, 1445–1453.
Lindsay, B. (1988), “Composite likelihood methods,” Contemporary Mathematics, 80, 221–239.
Padoan, S. A., and Bevilacqua, M. (2015), “Analysis of Random Fields Using CompRandFld,” Journal of Statistical Software, 63, 1–27.
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.
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.
Shaby, B., and Ruppert, D. (2012), “Tapered covariance: Bayesian estimation and asymptotics,” Journal of Computational and Graphical Statistics, 21, 433–452.
Stein, M. (2005), “Space-time covariance functions,” Journal of the American Statistical Association, 100, 310–321.
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.
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.
Varin, C., Reid, N., and Firth, D. (2011), “An overview of composite likelihood methods,” Statistica Sinica, 21, 5–42.
Varin, C., and Vidoni, P. (2005), “A note on composite likelihood inference and model selection,” Biometrika, 52, 519–528.
Wackernagel, H. (2003), Multivariate Geostatistics: An Introduction with Applications, 3rd edn, New York: Springer.
Wood, S. (2006), Generalized Additive Models: An Introduction with R, : Chapman and Hall CRC.
Zhang, H. (2007), “Maximum-likelihood estimation for multivariate spatial linear coregionalization models,” Environmetrics, 18, 125–139.
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.
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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
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DOI: https://doi.org/10.1007/s13253-016-0256-3