Bootstrap approaches for spatial data

  • Pilar García-Soidán
  • Raquel Menezes
  • Óscar Rubiños


Generation of replicates of the available data enables the researchers to solve different statistical problems, such as the estimation of standard errors, the inference of parameters or even the approximation of distribution functions. With this aim, Bootstrap approaches are suggested in the current work, specifically designed for their application to spatial data, as they take into account the dependence structure of the underlying random process. The key idea is to construct nonparametric distribution estimators, adapted to the spatial setting, which are distribution functions themselves, associated to discrete or continuous random variables. Then, the Bootstrap samples are obtained by drawing at random from the estimated distribution. Consistency of the suggested approaches will be proved by assuming stationarity from the random process or by relaxing the latter hypothesis to admit a deterministic trend. Numerical studies for simulated data and a real data set, obtained from environmental monitoring, are included to illustrate the application of the proposed Bootstrap methods.


Distribution estimation Resampling method Spatial data Stationarity Trend 

Mathematics Subject Classification

62G09 62M30 



The authors would like to thank the helpful suggestions and comments from the Reviewers. The authors are also grateful to Dr. K. J. Duncan-Barlow (University of Vigo) for her contribution in the language revision. The first and third authors acknowledge financial support from the Project TEC2011-28683-C02-02 of the Spanish Ministry of Science and Innovation and the Project CN2012/279 from the European Regional Development Fund and the Galician Regional Government (Xunta de Galicia). The second author’s work has been supported by the Project PTDC/MAT/112338/2009 (FEDER support included) of the Portuguese Ministry of Science, Technology and Higher Education.


  1. Bowman A, Hall P, Prvan T (1998) Bandwidth selection for the smoothing of distribution functions. Biometrika 85(4):799–808. doi: 10.1093/biomet/85.4.799 CrossRefGoogle Scholar
  2. Crujeiras RM, Fernández-Casal R, González-Manteiga W (2010) Goodness-of-fit tests for the spatial spectral density. Stoch Environ Res Risk Assess 24(1):67–79. doi: 10.1007/s00477-008-0300-0 CrossRefGoogle Scholar
  3. De Angelis D, Young GA (1992) Smoothing the bootstrap. Int Stat Rev 60(1):45–56CrossRefGoogle Scholar
  4. Efron B (1979) Bootstrap methods: another look at the Jackknife. Ann Stat 7(1):1–26. doi: 10.1214/aos/1176344552 CrossRefGoogle Scholar
  5. García-Soidán P (2007) Asymptotic normality of the Nadaraya–Watson semivariogram estimators. TEST 16(3):479–503. doi: 10.1007/s11749-006-0016-8 CrossRefGoogle Scholar
  6. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New YorkGoogle Scholar
  7. Govaerts B, Beck B, Lecoutre E, Le Bailly C, VandenEeckaut P (2005) From monitoring data to regional distributions: a practical methodology applied to water risk assessment. Environmetrics 16(2):109–127. doi: 10.1002/env.665 Google Scholar
  8. Hall P (1985) Resampling a coverage pattern. Stoch Process Appl 20(2):231–246. doi: 10.1016/0304-4149(85)90212-1 CrossRefGoogle Scholar
  9. Hall P (1992) The bootstrap and edgeworth expansion. Springer, New YorkGoogle Scholar
  10. Hall P, Maiti T (2006) On parametric bootstrap methods for small area prediction. J R Stat Soc B 68(2):221–238. doi: 10.1111/j.1467-9868.2006.00541.x CrossRefGoogle Scholar
  11. Hall P, Patil P (1994) Properties of nonparametric estimators of autocovariance for stationary random fields. Probab Theory Relat Fields 99(3):399–424. doi: 10.1007/BF01199899 CrossRefGoogle Scholar
  12. Hyun-Han K, Young-Il M (2006) Improvement of overtopping risk evaluations using probabilistic concepts for existing dams. Stoch Environ Res Risk Assess 20(4):223–237. doi: 10.1007/s00477-005-0017-2 CrossRefGoogle Scholar
  13. Iranpanah N, Mansourianb A, Tashayob B, Haghighic F (2011) Spatial semi-parametric bootstrap method for analysis of kriging predictor of random field. Procedia Environ Sci 3:81–86. doi: 10.1016/j.proenv.2011.02.015 CrossRefGoogle Scholar
  14. Lejeune M, Sarda P (1992) Smooth estimators of distribution and density functions. Comput Stat Data Anal 14:457–471. doi: 10.1016/0167-9473(92)90061-J CrossRefGoogle Scholar
  15. Li B, Genton M, Sherman M (2007) A nonparametric assessment of properties of space–time covariance functions. JASA 102(478):736–744. doi: 10.1198/016214507000000202 CrossRefGoogle Scholar
  16. Loh JM (2008) A valid and fast spatial Bootstrap for correlation functions. Astrophys J 681(1):726–734. doi: 10.1086/588631 CrossRefGoogle Scholar
  17. Maglione DS, Diblasi AM (2004) Exploring a valid model for the variogram of an isotropic spatial process. Stoch Environ Res Risk Assess 18(6):366–376. doi: 10.1007/s00477-003-0143-7 CrossRefGoogle Scholar
  18. Martins A, Figueira R, Sousa A, Sérgio C (2012) Spatio-temporal patterns of Cu contamination in mosses using geostatistical estimation. Environ Pollut 170:276–284. doi: 10.1016/j.envpol.2012.07.004 CrossRefGoogle Scholar
  19. Menezes R, García-Soidán P, Ferreira C (2010) Nonparametric spatial prediction under stochastic sampling design. J Nonparametr Stat 22(3):363–377. doi: 10.1080/10485250903094294 CrossRefGoogle Scholar
  20. Olea RA, Pardo-Igúzquiza E (2011) Generalized Bootstrap method for assessment of uncertainty in semivariogram inference. Math Geosci 43(2):203–228. doi: 10.1007/s11004-010-9269-6 CrossRefGoogle Scholar
  21. Politis DN, Romano JP, Wolf M (1999) Subsampling. Springer, BerlinGoogle Scholar
  22. Shapiro A, Botha JD (1991) Variogram fitting with a general class of conditionally nonnegative definite functions. Comput Stat Data Anal 11(1):87–96. doi: 10.1016/0167-9473(91)90055-7 CrossRefGoogle Scholar
  23. Silverman BW, Young GA (1987) The bootstrap: to smooth or not to smooth? Biometrika 74(3):469–479. doi: 10.1093/biomet/74.3.469 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Pilar García-Soidán
    • 1
  • Raquel Menezes
    • 2
  • Óscar Rubiños
    • 3
  1. 1.Department of Statistics and Operations ResearchUniversity of VigoVigoSpain
  2. 2.Department of Mathematics and ApplicationsUniversity of MinhoBragaPortugal
  3. 3.Department of Signal Theory and CommunicationsUniversity of VigoVigoSpain

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