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Fixed rank kriging for very large spatial data sets

Author

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  • Noel Cressie
  • Gardar Johannesson

Abstract

Spatial statistics for very large spatial data sets is challenging. The size of the data set, "n", causes problems in computing optimal spatial predictors such as kriging, since its computational cost is of order . In addition, a large data set is often defined on a large spatial domain, so the spatial process of interest typically exhibits non-stationary behaviour over that domain. A flexible family of non-stationary covariance functions is defined by using a set of basis functions that is fixed in number, which leads to a spatial prediction method that we call fixed rank kriging. Specifically, fixed rank kriging is kriging within this class of non-stationary covariance functions. It relies on computational simplifications when "n" is very large, for obtaining the spatial best linear unbiased predictor and its mean-squared prediction error for a hidden spatial process. A method based on minimizing a weighted Frobenius norm yields best estimators of the covariance function parameters, which are then substituted into the fixed rank kriging equations. The new methodology is applied to a very large data set of total column ozone data, observed over the entire globe, where "n" is of the order of hundreds of thousands. Copyright 2008 Royal Statistical Society.

Suggested Citation

  • Noel Cressie & Gardar Johannesson, 2008. "Fixed rank kriging for very large spatial data sets," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 209-226.
  • Handle: RePEc:bla:jorssb:v:70:y:2008:i:1:p:209-226
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9868.2007.00633.x
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    References listed on IDEAS

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    1. Steel, M.F.J., 1991. "Bayesian Inference in Time Series," Papers 9153, Tilburg - Center for Economic Research.
    2. Jose T.A.S. Ferreira & Mark F.J. Steel, 2004. "Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions," Econometrics 0403001, EconWPA.
    3. J. T. A. S. Ferreira & M. F. J. Steel, 2004. "On Describing Multivariate Skewness: A Directional Approach," Econometrics 0409010, EconWPA.
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    Cited by:

    1. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    2. Jonathan Bradley & Noel Cressie & Tao Shi, 2015. "Rejoinder on: Comparing and selecting spatial predictors using local criteria," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, pages 54-60.
    3. Jun Wang & Yang Wang & Hui Zeng, 2016. "A geostatistical approach to the change-of-support problem and variable-support data fusion in spatial analysis," Journal of Geographical Systems, Springer, pages 45-66.
    4. Hensley H Mariathas & Brajendra C Sutradhar, 2016. "Variable Family Size Based Spatial Moving Correlations Model," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, pages 1-38.
    5. repec:eee:csdana:v:117:y:2018:i:c:p:138-153 is not listed on IDEAS
    6. Andrew Finley & Sudipto Banerjee & Alan Gelfand, 2012. "Bayesian dynamic modeling for large space-time datasets using Gaussian predictive processes," Journal of Geographical Systems, Springer, pages 29-47.
    7. Giovanna Jona Lasinio & Gianluca Mastrantonio & Alessio Pollice, 2013. "Discussing the “big n problem”," Statistical Methods & Applications, Springer;Società Italiana di Statistica, pages 97-112.
    8. Hu Juan & Zhang Hao, 2015. "Numerical Methods of Karhunen–Loève Expansion for Spatial Data," Stochastics and Quality Control, De Gruyter, pages 49-58.
    9. Chandra, Hukum & Salvati, Nicola & Chambers, Ray & Tzavidis, Nikos, 2012. "Small area estimation under spatial nonstationarity," Computational Statistics & Data Analysis, Elsevier, pages 2875-2888.
    10. repec:jss:jstsof:v:072:i01 is not listed on IDEAS
    11. Jonathan Bradley & Noel Cressie & Tao Shi, 2015. "Comparing and selecting spatial predictors using local criteria," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, pages 1-28.
    12. Patrick Vetter & Wolfgang Schmid & Reimund Schwarze, 2013. "Efficient Approximation of the Spatial Covariance Function for Large Datasets - Analysis of Atmospheric CO2 Concentrations," Discussion Paper Series RECAP15 009, RECAP15, European University Viadrina, Frankfurt (Oder).
    13. Hossein Boojari & Majid Jafari Khaledi & Firoozeh Rivaz, 2016. "A non-homogeneous skew-Gaussian Bayesian spatial model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, pages 55-73.
    14. Bolin, David & Lindgren, Finn, 2013. "A comparison between Markov approximations and other methods for large spatial data sets," Computational Statistics & Data Analysis, Elsevier, pages 7-21.
    15. Furrer, Reinhard & Bachoc, François & Du, Juan, 2016. "Asymptotic properties of multivariate tapering for estimation and prediction," Journal of Multivariate Analysis, Elsevier, pages 177-191.
    16. Li, Yang & Zhu, Zhengyuan, 2016. "Modeling nonstationary covariance function with convolution on sphere," Computational Statistics & Data Analysis, Elsevier, pages 233-246.
    17. Huang, Chunfeng & Zhang, Haimeng & Robeson, Scott M., 2012. "A simplified representation of the covariance structure of axially symmetric processes on the sphere," Statistics & Probability Letters, Elsevier, pages 1346-1351.
    18. Sandy Burden & Noel Cressie & David G. Steel, 2015. "The SAR Model for Very Large Datasets: A Reduced Rank Approach," Econometrics, MDPI, Open Access Journal, vol. 3(2), pages 1-22, May.
    19. Ryan J. Parker & Brian J. Reich & Jo Eidsvik, 2016. "A Fused Lasso Approach to Nonstationary Spatial Covariance Estimation," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, pages 569-587.

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