A composite spatial predictor via local criteria under a misspecified model

Original Paper
  • 51 Downloads

Abstract

Spatial prediction and variable selection for the study area are both important issues in geostatistics. If spatially varying means exist among different subareas, globally fitting a spatial regression model for observations over the study area may be not suitable. To alleviate deviations from spatial model assumptions, this paper proposes a methodology to locally select variables for each subarea based on a locally empirical conditional Akaike information criterion. In this situation, the global spatial dependence of observations is considered and the local characteristics of each subarea are also identified. It results in a composite spatial predictor which provides a more accurate spatial prediction for the response variables of interest in terms of the mean squared prediction errors. Further, the corresponding prediction variance is also evaluated based on a resampling method. Statistical inferences of the proposed methodology are justified both theoretically and numerically. Finally, an application of a mercury data set for lakes in Maine, USA is analyzed for illustration.

Keywords

Geostatistics Information criterion Prediction variance Resampling Squared prediction error 

Notes

Acknowledgements

We thank the Editor, an associate editor, and two anonymous referees for their helpful comments and suggestions. This work was supported by the Ministry of Science and Technology of Taiwan under Grant MOST 104-2118-M-018-002-MY2.

References

  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov V, Csáki F (eds) International symposium on information theory. Akademiai Kiádo, Budapest, pp 267–281Google Scholar
  2. Assuncão R (2003) Space varying coefficient models for small area data. Environmetrics 14:453–473CrossRefGoogle Scholar
  3. Borra S, Di Ciaccio A (2010) Measuring the prediction error: a comparison of cross-validation, bootstrap and covariance penalty methods. Comput Stat Data Anal 54:2976–2989CrossRefGoogle Scholar
  4. Bradley JR, Cressie N, Shi T (2015) Comparing and selecting spatial predictors using local criteria. Test 24:1–28CrossRefGoogle Scholar
  5. Chen CS, Huang HC (2012) Geostatistical model averaging based on conditional information criteria. Environ Ecol Stat 19:23–35CrossRefGoogle Scholar
  6. Chilés JP, Delfinder JP (1999) Geostatistics: modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  7. Cressie N, Johannesson G (2008) Fixed rank kriging for very large data sets. J R Stat Soc Ser B 70:209–226CrossRefGoogle Scholar
  8. Cressie N, Lahiri SN (1996) Asymptotics for REML estimation of spatial covariance parameters. J Stat Plan Inference 50:327–341CrossRefGoogle Scholar
  9. Davison A, Hinkley D (1997) Bootstrap methods and their application. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. Efron B (2004) The estimation of prediction error: covariance penalties and cross-validation. J Am Stat Assoc 99:619–632CrossRefGoogle Scholar
  11. Efron B (2014) Estimation and accuracy after model selection. J Am Stat Assoc 109:991–1007CrossRefGoogle Scholar
  12. Fahrmeir L, Kneib T, Lang S (2004) Penalized structured additive regression for space-time data: a Bayesian perspective. Stat Sin 14:731–761Google Scholar
  13. Fouedjio F (2016) Second-order non-stationary modeling approaches for univariate geostatistical data. Stoch Environ Res Risk Assess. doi: 10.1007/s00477-016-1274-y Google Scholar
  14. Furrer R, Genton MG, Nychka D (2006) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15:502–523CrossRefGoogle Scholar
  15. García-Soidán P, Menezes R, Rubiños Ó (2014) Bootstrap approaches for spatial data. Stoch Environ Res Risk Assess 28:1207–1219CrossRefGoogle Scholar
  16. Ghosh D, Yuan Z (2009) An improved model averaging scheme for logistic regression. J Multivar Anal 100:1670–1681CrossRefGoogle Scholar
  17. Hoeting JA, Davis RA, Merton AA, Thompson SE (2006) Model selection for geostatistical models. Ecol Appl 16:87–98CrossRefGoogle Scholar
  18. Jiang W, Simon R (2007) A comparison of bootstrap methods and an adjusted bootstrap approach for estimating the prediction error in microarray classification. Stat Med 26:5320–5334CrossRefGoogle Scholar
  19. Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103:1545–1555CrossRefGoogle Scholar
  20. Lloyd CD (2011) Local models for spatial analysis, 2nd edn. CRC Press, Boca RatonGoogle Scholar
  21. Matérn B (2013) Spatial variation. Springer, BerlinGoogle Scholar
  22. McGilchrist CA (1989) Bias of ML and REML estimators in regression models with ARMA errors. J Stat Comput Simul 32:127–136CrossRefGoogle Scholar
  23. Paciorek C, Schervish M (2006) Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17:483–506CrossRefGoogle Scholar
  24. Patterson HD, Thompson R (1971) Recovery of inter-block information when block sizes are unequal. Biometrika 58:545–554CrossRefGoogle Scholar
  25. Peck R, Haugh LD, Goodman A, (eds) (1998) Statistical case studies: a collaboration between academe and industry. In: ASA-SIAM series on statistics and applied probability 3 and 4Google Scholar
  26. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464CrossRefGoogle Scholar
  27. Shen X, Huang HC, Ye J (2004) Adaptive model selection and assessment for exponential family models. Technometrics 46:306–317CrossRefGoogle Scholar
  28. Tutmez B, Kaymak U, Tercan AE (2012) Local spatial regression models: a comparative analysis on soil contamination. Stoch Environ Res Risk Assess 26:1013–1023CrossRefGoogle Scholar
  29. Vaida F, Blanchard S (2005) Conditional Akaike information for mixed-effects models. Biometrika 92:351–370CrossRefGoogle Scholar
  30. Yang HD, Chen CS (2017) On estimation and prediction of geostatistical regression models via a corrected Stein’s unbiased risk estimator. Environmetrics 28:e2424. doi: 10.1002/env.2424 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Statistics and Information ScienceNational Changhua University of EducationChanghuaTaiwan

Personalised recommendations