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Spectral methods for nonstationary spatial processes


  • Montserrat Fuentes


We propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process. We also study the asymptotic properties of the proposed estimators via shrinking asymptotics, assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound. With this type of asymptotic model we can uniquely determine the spectral density, avoiding the aliasing problem. We also present a new class of nonstationary processes, based on a convolution of local stationary processes. This model has the advantage that the model is simultaneously defined everywhere, unlike 'moving window' approaches, but it retains the attractive property that, locally in small regions, it behaves like a stationary spatial process. Applications include the spatial analysis and modelling of air pollution data provided by the US Environmental Protection Agency. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • Montserrat Fuentes, 2002. "Spectral methods for nonstationary spatial processes," Biometrika, Biometrika Trust, vol. 89(1), pages 197-210, March.
  • Handle: RePEc:oup:biomet:v:89:y:2002:i:1:p:197-210

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    Cited by:

    1. Fernández-Avilés, G & Montero, JM & Mateu, J, 2011. "Mathematical Genesis of the Spatio-Temporal Covariance Functions," MPRA Paper 35874, University Library of Munich, Germany.
    2. Montserrat Fuentes & Peter Guttorp & Peter Challenor, 2003. "Statistical Assessment of Numerical Models," International Statistical Review, International Statistical Institute, vol. 71(2), pages 201-221, August.
    3. Joaquim Henriques Vianna Neto & Alexandra M. Schmidt & Peter Guttorp, 2014. "Accounting for spatially varying directional effects in spatial covariance structures," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 63(1), pages 103-122, January.
    4. repec:bla:jorssb:v:79:y:2017:i:1:p:95-123 is not listed on IDEAS
    5. repec:eee:reensy:v:113:y:2013:i:c:p:30-41 is not listed on IDEAS
    6. Christopher K. Wikle, 2003. "Hierarchical Models in Environmental Science," International Statistical Review, International Statistical Institute, vol. 71(2), pages 181-199, August.
    7. Alan Gelfand & Alexandra Schmidt & Sudipto Banerjee & C. Sirmans, 2004. "Nonstationary multivariate process modeling through spatially varying coregionalization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 263-312, December.
    8. Tatiyana V. Apanasovich & David Ruppert & Joanne R. Lupton & Natasa Popovic & Nancy D. Turner & Robert S. Chapkin & Raymond J. Carroll, 2008. "Aberrant Crypt Foci and Semiparametric Modeling of Correlated Binary Data," Biometrics, The International Biometric Society, vol. 64(2), pages 490-500, June.
    9. Peter J. Brockwell & Yasumasa Matsuda, 2015. "Levy-driven CARMA Random Fields on Rn," TERG Discussion Papers 339, Graduate School of Economics and Management, Tohoku University.
    10. Kleiber, William & Nychka, Douglas, 2012. "Nonstationary modeling for multivariate spatial processes," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 76-91.
    11. repec:bla:jorssb:v:79:y:2017:i:3:p:833-857 is not listed on IDEAS
    12. Chiranjit Mukherjee & Prasad Kasibhatla & Mike West, 2014. "Spatially varying SAR models and Bayesian inference for high-resolution lattice data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(3), pages 473-494, June.
    13. Alexandre Rodrigues & Peter J. Diggle, 2010. "A Class of Convolution-Based Models for Spatio-Temporal Processes with Non-Separable Covariance Structure," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 553-567.
    14. Fuentes, Montserrat, 2005. "A formal test for nonstationarity of spatial stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 30-54, September.
    15. repec:bla:jorssc:v:66:y:2017:i:5:p:919-939 is not listed on IDEAS
    16. repec:bla:biomet:v:73:y:2017:i:3:p:759-768 is not listed on IDEAS
    17. Martin Tingley & Benjamin Shaby, 2015. "Comments 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, vol. 24(1), pages 47-53, March.
    18. Kelejian, Harry H. & Prucha, Ingmar R., 2007. "HAC estimation in a spatial framework," Journal of Econometrics, Elsevier, vol. 140(1), pages 131-154, September.
    19. repec:bla:jtsera:v:38:y:2017:i:2:p:204-224 is not listed on IDEAS
    20. Li, Yang & Zhu, Zhengyuan, 2016. "Modeling nonstationary covariance function with convolution on sphere," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 233-246.
    21. Lim, Chae Young & Stein, Michael, 2008. "Properties of spatial cross-periodograms using fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1962-1984, October.
    22. Whitcher, Brandon, 2006. "Wavelet-based bootstrapping of spatial patterns on a finite lattice," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2399-2421, May.
    23. 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, vol. 21(3), pages 569-587, September.

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