Exploration and Inference in Spatial Extremes Using Empirical Basis Functions

  • Samuel A. MorrisEmail author
  • Brian J. Reich
  • Emeric Thibaud


Statistical methods for inference on spatial extremes of large datasets are yet to be developed. Motivated by standard dimension reduction techniques used in spatial statistics, we propose an approach based on empirical basis functions to explore and model spatial extremal dependence. Based on a low-rank max-stable model, we propose a data-driven approach to estimate meaningful basis functions using empirical pairwise extremal coefficients. These spatial empirical basis functions can be used to visualize the main trends in extremal dependence. In addition to exploratory analysis, we describe how these functions can be used in a Bayesian hierarchical model to model spatial extremes of large datasets. We illustrate our methods on extreme precipitations in eastern USA.

Supplementary materials accompanying this paper appear online


Dimension reduction Max-stable process Non-stationary data analysis 



The authors acknowledge Dan Cooley and Michael Wehner for their helpful suggestions on the manuscript. The authors’ work was partially supported by Grants from the Department of the Interior (14-1-04-9), National Institutes of Health (R21ES022795-01A1), the US Environmental Protection Agency (R835228), the National Science Foundation (1107046). The calculations have been performed using the facilities of the Scientific IT and Application Support Center of EPFL.

Supplementary material

Supplementary material 1 (RData 174 KB)


  1. Bernard, E., Naveau, P., Vrac, M., and Mestre, O. (2013). Clustering of Maxima: Spatial Dependencies among Heavy Rainfall in France. Journal of Climate, 26(20):7929–7937.CrossRefGoogle Scholar
  2. Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer, London.CrossRefzbMATHGoogle Scholar
  3. Cooley, D., Naveau, P., and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Bertail, P., Soulier, P., and Doukhan, P., editors, Dependence in Probability and Statistics, volume 187 of Lecture Notes in Statistics, chapter Variograms, pages 373–390. Springer New York, New York, NY.Google Scholar
  4. Davison, A. C., Huser, R., and Thibaud, E. (2013). Geostatistics of Dependent and Asymptotically Independent Extremes. Mathematical Geosciences, 45(5):511–529.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Davison, A. C., Padoan, S. A., and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statistical Science, 27(2):161–186.MathSciNetCrossRefzbMATHGoogle Scholar
  6. de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer.Google Scholar
  7. Dey, D. K. and Yan, J. (2015). Extreme Value Modeling and Risk Analysis: Methods and Applications. Chapman and Hall/CRC, Boca Raton.CrossRefGoogle Scholar
  8. Einmahl, J. H. J., Kiriliouk, A., and Segers, J. (2016). A continuous updating weighted least squares estimator of tail dependence in high dimensions. arXiv:1601.04826.
  9. Engelke, S., Malinowski, A., Kabluchko, Z., and Schlather, M. (2015). Estimation of Hüsler-Reiss distributions and Brown-Resnick processes. Journal of the Royal Statistical Society. Journal of the Royal Statistical Society Series B: Statistical Methodology, 77:239–265.MathSciNetCrossRefGoogle Scholar
  10. Everitt, B. and Hothorn, T. (2008). Principal components analysis. In An Introduction to Applied Multivariate Analysis with R, pages 21–54. Springer New York, New York, NY.Google Scholar
  11. Fougères, A.-L., Mercadier, C., and Nolan, J. P. (2013). Dense classes of multivariate extreme value distributions. Journal of Multivariate Analysis, 116:109–129.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fougères, A.-L., Nolan, J. P., and Rootzén, H. (2009). Models for Dependent Extremes Using Stable Mixtures. Scandinavian Journal of Statistics, 36(1):42–59.MathSciNetzbMATHGoogle Scholar
  13. Hannachi, A., Jolliffe, I. T., and Stephenson, D. B. (2007). Empirical orthogonal functions and related techniques in atmospheric science: A review. International Journal of Climatology, 27(9):1119–1152.CrossRefGoogle Scholar
  14. Huser, R. and Davison, A. C. (2014). Space-time modelling of extreme events. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2):439–461.MathSciNetCrossRefGoogle Scholar
  15. Kabluchko, Z., Schlather, M., and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Annals of Probability, 37(5):2042–2065.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lee, D. D. and Seung, S. H. (1999). Learning the parts of objects by non-negative matrix factorizations. Nature, 401:788 – 791.CrossRefzbMATHGoogle Scholar
  17. Mairal, J., Bach, F., and Ponce, J. (2014). Sparse modeling for image and vision processing. Foundations and Trends in Computer Graphics and Vision, 8:85 – 283.CrossRefzbMATHGoogle Scholar
  18. Morris, S. A., Reich, B. J., Thibaud, E., and Cooley, D. (2017). A space-time skew-t model for threshold exceedances. Biometrics, 73(3):749–758.MathSciNetCrossRefGoogle Scholar
  19. Nychka, D., Furrer, R., and Sain, S. (2015). fields: Tools for Spatial Data. R package version 8.2-1.Google Scholar
  20. Padoan, S. A., Ribatet, M., and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. Journal of the American Statistical Association, 105(489):263–277.MathSciNetCrossRefzbMATHGoogle Scholar
  21. R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.Google Scholar
  22. Reich, B. J. and Shaby, B. A. (2012). A hierarchical max-stable spatial model for extreme precipitation. The Annals of Applied Statistics, 6(4):1430–1451.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Reich, B. J., Shaby, B. A., and Cooley, D. (2014). A Hierarchical Model for Serially-Dependent Extremes: A Study of Heat Waves in the Western US. Journal of Agricultural, Biological, and Environmental Statistics, 19(1):119–135.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Ribatet, M. (2015). SpatialExtremes: Modelling Spatial Extremes. R package version 2.0-2.Google Scholar
  25. Schlather, M. (2002). Models for Stationary Max-Stable Random Fields. Extremes, 5(1):33–44.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika, 90(1):139–156.MathSciNetCrossRefzbMATHGoogle Scholar
  27. Shaby, B. A. and Reich, B. J. (2012). Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland. Environmetrics, 23(8):638–648.MathSciNetCrossRefGoogle Scholar
  28. Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript, University of Surrey, Guildford GU2 5XH, England.
  29. Stephenson, A. G. (2009). High-dimensional parametric modelling of multivariate extreme events. Australian & New Zealand Journal of Statistics, 51(1):77–88.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Stephenson, A. G., Shaby, B. A., Reich, B. J., and Sullivan, A. L. (2015). Estimating Spatially Varying Severity Thresholds of a Forest Fire Danger Rating System Using Max-Stable Extreme-Event Modeling. Journal of Applied Meteorology and Climatology, 54(2):395–407.CrossRefGoogle Scholar
  31. Thibaud, E., Aalto, J., Cooley, D. S., Davison, A. C., and Heikkinen, J. (2016). Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures. Annals of Applied Statistics, 10(4):2303–2324.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Thibaud, E., Mutzner, R., and Davison, A. C. (2013). Threshold modeling of extreme spatial rainfall. Water Resources Research, 49(8):4633–4644.CrossRefGoogle Scholar
  33. Thibaud, E. and Opitz, T. (2015). Efficient inference and simulation for elliptical Pareto processes. Biometrika, 102(4):855–870.MathSciNetCrossRefzbMATHGoogle Scholar
  34. Wadsworth, J. L. and Tawn, J. A. (2014). Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika, 101(1):1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Wang, Y. and Stoev, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Advances in Applied Probability, 43(2):461–483.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Wehner, M. F. (2013). Very extreme seasonal precipitation in the NARCCAP ensemble: model performance and projections. Climate Dynamics, 40(1):59–80.MathSciNetCrossRefGoogle Scholar

Copyright information

© International Biometric Society 2019

Authors and Affiliations

  1. 1.North Carolina State UniversityRaleighUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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