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Covariance Tapering for Prediction of Large Spatial Data Sets in Transformed Random Fields

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  • Toshihiro Hirano

    (Graduate School of Economics, University of Tokyo)

  • Yoshihiro Yajima

    (Faculty of Ecocnomics, University of Tokyo)

Abstract

The best linear unbiased predictor (BLUP) is called a kriging predictor and has been widely used to interpolate a spatially correlated random process in scientific areas such as geostatistics. The BLUP is identical with the conditional expectation if an underlying random field is Gaussian and consequently is the optimal predictor in the mean squared error (MSE) sense. However, if an original data takes a nonnegative value or has a skewed distribution, we frequently apply a nonlinear transformation to it to get a data which is nearer Gaussian. Then the optimality of the BLUP for the original data is unclear because it is not Gaussian. Moreover, in many cases, data sets in spatial problems are often so large that a kriging predictor is impractically time-consuming. To reduce the computational complexity, covariance tapering has been developed by Furrer et al. (2006) for large spatial data sets. In this paper we consider covariance tapering in a class of transformed Gaussian models for random elds and show that the BLUP using covariance tapering, the BLUP and the optimal predictor are asymptotically equivalent in the MSE sense if the underlying Gaussian random eld has the Matérn covariance function. This is an extension of Furrer et al. (2006). Monte Carlo simulations support theoretical results.

Suggested Citation

  • Toshihiro Hirano & Yoshihiro Yajima, 2011. "Covariance Tapering for Prediction of Large Spatial Data Sets in Transformed Random Fields," CIRJE F-Series CIRJE-F-823, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2011cf823
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    References listed on IDEAS

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    1. Johns C.J. & Nychka D. & Kittel T.G.F. & Daly C., 2003. "Infilling Sparse Records of Spatial Fields," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 796-806, January.
    2. Kaufman, Cari G. & Schervish, Mark J. & Nychka, Douglas W., 2008. "Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1545-1555.
    3. Toshihiro Hirano & Yoshihiro Yajima, 2013. "Covariance tapering for prediction of large spatial data sets in transformed random fields," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(5), pages 913-939, October.
    4. Furrer, Reinhard & Sain, Stephan R., 2010. "spam: A Sparse Matrix R Package with Emphasis on MCMC Methods for Gaussian Markov Random Fields," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 36(i10).
    5. Yakowitz, S. J. & Szidarovszky, F., 1985. "A comparison of kriging with nonparametric regression methods," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 21-53, February.
    6. Victor De Oliveira, 2006. "On Optimal Point and Block Prediction in Log‐Gaussian Random Fields," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(3), pages 523-540, September.
    7. Stein, Michael L., 1993. "A simple condition for asymptotic optimality of linear predictions of random fields," Statistics & Probability Letters, Elsevier, vol. 17(5), pages 399-404, August.
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    Cited by:

    1. Toshihiro Hirano & Yoshihiro Yajima, 2013. "Covariance tapering for prediction of large spatial data sets in transformed random fields," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(5), pages 913-939, October.
    2. Matthew J. Heaton & Abhirup Datta & Andrew O. Finley & Reinhard Furrer & Joseph Guinness & Rajarshi Guhaniyogi & Florian Gerber & Robert B. Gramacy & Dorit Hammerling & Matthias Katzfuss & Finn Lindgr, 2019. "A Case Study Competition Among Methods for Analyzing Large Spatial Data," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(3), pages 398-425, September.

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