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Non-Gaussian Covariate-Dependent Spatial Measurement Error Model for Analyzing Big Spatial Data

  • Vahid TadayonEmail author
  • Abdolrahman Rasekh
Article
  • 43 Downloads

Abstract

Spatial models based on the Gaussian distribution have been widely used in environmental sciences. However, real data could be highly non-Gaussian and may show heavy tails features. Moreover, as in any type of statistical models, in spatial statistical models, it is commonly assumed that the covariates are observed without errors. Nonetheless, for various reasons such as measurement techniques or instruments used, measurement error (ME) can be present in the covariates of interest. This article concentrates on modeling heavy-tailed geostatistical data using a more flexible class of ME models. One novelty of this article is to allow the spatial covariance structure to depend on ME. For this purpose, we adopt a Bayesian modeling approach and utilize Markov chain Monte Carlo techniques and data augmentations to carry out the inference. However, when the number of observations is large, statistical inference is computationally burdensome, since the covariance matrix needs to be inverted at each iteration. As another novelty, we use a prediction-oriented Bayesian site selection scheme to tackle this difficulty. The proposed approach is illustrated with a simulation study and an application to nitrate concentration data. Supplementary materials accompanying this paper appear online.

Keywords

Bayesian site selection Covariate-dependent spatial covariance function Gaussian log-Gaussian spatial measurement error model Spatial heteroscedasticity 

Notes

Acknowledgements

The Associate Editor and two referees are gratefully acknowledged. Their precise comments and constructive suggestions have clearly improved the manuscript.

Supplementary material

13253_2018_341_MOESM1_ESM.zip (69 kb)
Supplementary Materials The supplementary materials contain R codes and corresponding “ReadMe” files for the simulation and real data application conducted in this paper. (zip 69 KB)

References

  1. S.E. Alexeeff, R.J. Carroll, and B. Coull. Spatial measurement error and correction by spatial SIMEX in linear regression models when using predicted air pollution exposures. Biostatistics, 17(2):377–389, 2016.MathSciNetCrossRefGoogle Scholar
  2. R. Bastian and D. Murray. Guidelines for water reuse. US EPA Office of Research and Development, Washington, DC, EPA/600/R-12/618, 2012.Google Scholar
  3. R.S. Bueno, T.C.O. Fonseca, and A.M. Schmidt. Accounting for covariate information in the scale component of spatio-temporal mixing models. Spatial Statistics, 22:196–218, 2017.MathSciNetCrossRefGoogle Scholar
  4. E.K. Choi and Y.P. Kim. Effects of aerosol hygroscopicity on fine particle mass concentration and light extinction coefficient at Seoul and Gosan in Korea. Asian Journal of Atmospheric Environment, 4(1):55–61, 2010.CrossRefGoogle Scholar
  5. T.C.O Fonseca and M.F.J. Steel. Non-Gaussian spatio-temporal modelling through scale mixing. Biometrika, 98(4):761–774, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  6. W.A. Fuller. Measurement Error Models. John Wiley & Sons, 2009.Google Scholar
  7. A. Gelman and D.B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, pages 457–472, 1992.Google Scholar
  8. P.J. Green. Reversible jump markov chain monte carlo computation and Bayesian model determination. Biometrika, 82(4):711–732, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  9. R. Haining. Spatial Data Analysis in the Social and Environmental Sciences. Cambridge University Press, 1993.Google Scholar
  10. M.J. Heaton, A. Datta, A. Finley, R. Furrer, R. Guhaniyogi, F. Gerber, R.B. Gramacy, D. Hammerling, M. Katzfuss, F. Lindgren, et al. Methods for analyzing large spatial data: A review and comparison. arXiv preprint arXiv:1710.05013, 2017.
  11. M.H. Huque, H.D. Bondell, and L. Ryan. On the impact of covariate measurement error on spatial regression modelling. Environmetrics, 25(8):560–570, 2014.MathSciNetCrossRefGoogle Scholar
  12. M.H. Huque, H.D. Bondell, R.J. Carroll, and L.M. Ryan. Spatial regression with covariate measurement error: A semiparametric approach. Biometrics, 72(3):678–686, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  13. R. Ingebrigtsen, F. Lindgren, and I. Steinsland. Spatial models with explanatory variables in the dependence structure. Spatial Statistics, 8:20–38, 2014.MathSciNetCrossRefGoogle Scholar
  14. J.C. Jarvis, M.G. Hastings, E.J. Steig, and S.A. Kunasek. Isotopic ratios in gas-phase HNO\(_3\) and snow nitrate at Summit, Greenland. Journal of Geophysical Research: Atmospheres, 114(D17):1–14, 2009.CrossRefGoogle Scholar
  15. H.M. Kim and B.K. Mallick. A Bayesian prediction using the skew Gaussian distribution. Journal of Statistical Planning and Inference, 120(1-2):85–101, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Y. Li, H. Tang, and X. Lin. Spatial linear mixed models with covariate measurement errors. Statistica Sinica, 19(3):1077, 2009.MathSciNetzbMATHGoogle Scholar
  17. J.H. V. Neto, A.M. Schmidt, and P. Guttorp. Accounting for spatially varying directional effects in spatial covariance structures. Journal of the Royal Statistical Society: Series C (Applied Statistics), 63(1):103–122, 2014.MathSciNetCrossRefGoogle Scholar
  18. B.T. Nolan and J.D. Stoner. Nutrients in groundwaters of the conterminous United States, 1992- 1995. Environmental Science & Technology, 34(7):1156–1165, 2000.CrossRefGoogle Scholar
  19. S.P. Opsahl, M. Musgrove, and R.N. Slattery. New insights into nitrate dynamics in a karst groundwater system gained from in situ high-frequency optical sensor measurements. Journal of hydrology, 546:179–188, 2017.CrossRefGoogle Scholar
  20. M.B. Palacios and M.F.J. Steel. Non-Gaussian Bayesian geostatistical modeling. Journal of the American Statistical Association, 101(474):604–618, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  21. J. Park and F. Liang. A prediction-oriented Bayesian site selection approach for large spatial data. Journal of Statistical Research, 47(1):11–30, 2015.MathSciNetGoogle Scholar
  22. B.J. Reich, J. Eidsvik, M. Guindani, A.J. Nail, and A.M. Schmidt. A class of covariate-dependent spatio-temporal covariance functions for the analysis of daily ozone concentration. The Annals of Applied Statistics, 5(4):2265, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  23. A. Sathasivan, I. Fisher, and T. Tam. Onset of severe nitrification in mildly nitrifying chloraminated bulk waters and its relation to biostability. Water research, 42(14):3623–3632, 2008.CrossRefGoogle Scholar
  24. A.M. Schmidt, P. Guttorp, and A. O’Hagan. Considering covariates in the covariance structure of spatial processes. Environmetrics, 22(4):487–500, 2011.MathSciNetCrossRefGoogle Scholar
  25. M.F.J Steel and M. Fuentes. Non-Gaussian and non-parametric models for continuous spatial data. Handbook of Spatial Statistics, pages 149–167, 2010.Google Scholar
  26. V. Tadayon. Bayesian analysis of censored spatial data based on a non-gaussian model. Journal of Statistical Research of Iran, 13(2):155–180, 2017.CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2018

Authors and Affiliations

  1. 1.Department of StatisticsShahid Chamran University of AhvazAhvazIran

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