The Annals of Applied Statistics

Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping

David Bolin and Finn Lindgren

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A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions. Nonstationary covariance models are obtained by spatially varying the parameters in the SPDEs, and the model parameters are estimated using direct numerical optimization, which is more efficient than standard Markov Chain Monte Carlo procedures. The model class is used to estimate daily ozone maps using a large data set of spatially irregular global total column ozone data.

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Ann. Appl. Stat., Volume 5, Number 1 (2011), 523-550.

First available in Project Euclid: 21 March 2011

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Nested SPDEs Matérn covariances nonstationary covariances total column ozone data


Bolin, David; Lindgren, Finn. Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping. Ann. Appl. Stat. 5 (2011), no. 1, 523--550. doi:10.1214/10-AOAS383.

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  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.
  • Bolin, D. and Lindgren, F. (2009). Wavelet Markov approximations as efficient alternatives to tapering and convolution fields (submitted). Preprints in Math. Sci. 2009:13, Lund Univ.
  • Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 209–226.
  • Gelman, A., Roberts, G. O. and Gilks, W. R. (1996). Efficient Metropolis jumping rules. Bayesian Stat. 5 599–607.
  • Gneiting, T. (1998). Simple tests for the validity of correlation function models on the circle. Statist. Probab. Lett. 39 119–122.
  • Hastie, T., Tibshirani, R. and Friedman, J. H. (2001). The Elements of Statistical Learning. Springer, New York.
  • Hill, E. L. (1954). The theory of vector spherical harmonics. Amer. J. Phys. 22 211–214.
  • Jun, M. and Stein, M. L. (2007). An approach to producing space–time covariance functions on spheres. Technometrics 49 468–479.
  • Jun, M. and Stein, M. L. (2008). Nonstationary covariance models for global data. Ann. Appl. Statist. 2 1271–1289.
  • Lindgren, F., Rue, H. and Lindström, J. (2010). An explicit link between Gaussian fields and Gaussian Markov random fields, The SPDE approach (submitted). Preprints in Math. Sci. 2010:3, Lund Univ.
  • Lindström, J. and Lindgren, F. (2008). A Gaussian Markov random field model for total yearly precipitation over the African Sahel. Preprints in Math. Sci. 2008:8, Lund Univ.
  • Matérn, B. (1960). Spatial Variation. Meddelanden från statens skogsforskningsinstitut, Stockholm.
  • McPeters, R. D., Bhartia, P. K., Krueger, A. J., Herman, J. R., Schlesinger, B., Wellemeyer, C. G., Seftor, C. J., Jaross, G., Taylor, S. L., Swissler, T., Torres, O., Labow, G., Byerly, W. and Cebula, R. P. (1996). Nimbus-7 Total Ozone Mapping Spectrometer (TOMS) data products user’s guide. NASA Reference Publication 1384.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC, Boca Raton, FL.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested laplace approximations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319–392.
  • Stein, M. L. (2007). Spatial variation of total column ozone on a global scale. Ann. Appl. Statist. 1 191–210.
  • Thomas, G. B. and Finney, R. L. (1995). Calculus and Analytic Geometry, 9 ed. Addison Wesley, New York.
  • Whittle, P. (1963). Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40 974–994.
  • Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions 1. Springer, New York.