Non-Stationary Dependence Structures for Spatial Extremes


Max-stable processes are natural models for spatial extremes because they provide suitable asymptotic approximations to the distribution of maxima of random fields. In the recent past, several parametric families of stationary max-stable models have been developed, and fitted to various types of data. However, a recurrent problem is the modeling of non-stationarity. In this paper, we develop non-stationary max-stable dependence structures in which covariates can be easily incorporated. Inference is performed using pairwise likelihoods, and its performance is assessed by an extensive simulation study based on a non-stationary locally isotropic extremal t model. Evidence that unknown parameters are well estimated is provided, and estimation of spatial return level curves is discussed. The methodology is demonstrated with temperature maxima recorded over a complex topography. Models are shown to satisfactorily capture extremal dependence.

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  1. Anderes, E. B. and Stein, M. L. (2008) Estimating Deformations of Isotropic Gaussian Random Fields on the Plane. Annals of Statistics 36(2), 719–741.

  2. ——– (2011) Local Likelihood Estimation for Nonstationary Random Fields. Journal of Multivariate Analysis 102(3), 506–520.

  3. Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes: Theory and Applications. Chichester: Wiley. ISBN 9780471976479.

  4. Blanchet, J. and Davison, A. C. (2011) Spatial Modelling of Extreme Snow Depth. Annals of Applied Statistics 5(3), 1699–1725.

  5. Bornn, L., Shaddick, G. and Zidek, J. V. (2012) Modeling Nonstationary Processes Through Dimension Expansion. Journal of the American Statistical Association 107(497), 281–289.

  6. Brown, B. M. and Resnick, S. I. (1977) Extreme Values of Independent Stochastic Processes. Journal of Applied Probability 14(4), 732–739.

  7. Capéraà, P., Fougères, A.-L. and Genest, C. (1997) A Nonparametric Estimation Procedure for Bivariate Extreme Value Copulas. Biometrika 84(3), 567–577.

  8. de Carvalho, M. (2015) Statistics of Extremes: Challenges and Opportunities. In Extreme Events in Finance, ed. F. Longin. New York: Wiley.

  9. de Carvalho, M. and Davison, A. C. (2014) Spectral Density Ratio Models for Multivariate Extremes. Journal of the American Statistical Association 109(506), 764–776.

  10. Castro, D., de Carvalho, M. and Wadsworth, J. L. (2015) Time-Varying Extreme Value Dependence With Application to Leading European Stock Markets. Submitted.

  11. Castruccio, S. and Genton, M. G. (2016) Compressing an Ensemble with Statistical Models: An Algorithm for Global 3D Spatio-Temporal Temperature. Technometrics To appear.

  12. Castruccio, S., Huser, R. and Genton, M. G. (2016) High-Order Composite Likelihood Inference for Max-Stable Distributions and Processes. Journal of Computational and Graphical Statistics To appear.

  13. Castruccio, S. and Stein, M. L. (2013) Global Space-Time Models for Climate Ensembles. Annals of Applied Statistics 7(3), 1593–1611.

  14. Chavez-Demoulin, V. and Davison, A. C. (2005) Generalized Additive Modelling of Sample Extremes. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54(1), 207–222.

  15. Cooley, D. S., Cisewski, J., Erhardt, R. J., Jeon, S., Mannshardt-Shamseldin, E. C., Omolo, B. O. and Sun, Y. (2012) A Survey of Spatial Extremes: Measuring Spatial Dependence and Modeling Spatial Effects. REVSTAT 10(1), 135–165.

  16. Cooley, D. S., Naveau, P. and Nychka, D. (2007) Bayesian Spatial Modeling of Extreme Precipitation Return Levels. Journal of the American Statistical Association 102(479), 824–840.

  17. Cressie, N. A. C. (1993) Statistics for Spatial Data. New York: Wiley.

    Google Scholar 

  18. Cressie, N. A. C. and Wikle, C. K. (2011) Statistics for Spatio-Temporal Data. Hoboken: Wiley. ISBN 9780471692744.

    Google Scholar 

  19. Davis, R. A., Klüppelberg, C. and Steinkohl, C. (2013) Statistical Inference for Max-Stable Processes in Space and Time. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 75(5), 791–819.

  20. Davison, A. C. and Gholamrezaee, M. M. (2012) Geostatistics of Extremes. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences 468(2138), 581–608.

  21. Davison, A. C. and Huser, R. (2015) Statistics of Extremes. Annual Review of Statistics and its Application 2, 203–235.

  22. Davison, A. C., Huser, R. and Thibaud, E. (2013) Geostatistics of Dependent and Asymptotically Independent Extremes. Mathematical Geosciences 45(5), 511–529.

  23. Davison, A. C., Padoan, S. and Ribatet, M. (2012) Statistical Modelling of Spatial Extremes (with Discussion). Statistical Science 27(2), 161–186.

  24. Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2015) Estimation of Huesler–Reiss distributions and Brown–Resnick processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77(1), 239–265.

  25. Fuentes, M. (2001) A High Frequency Kriging Approach for Non-Stationary Environmental Processes. Environmetrics 12(5), 469–483.

  26. Genton, M. G., Ma, Y. and Sang, H. (2011) On the Likelihood Function of Gaussian Max-Stable Processes. Biometrika 98(2), 481–488.

  27. de Haan, L. (1984) A Spectral Representation for Max-Stable Processes. Annals of Probability 12(4), 1194–1204.

  28. de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction. New York: Springer. ISBN 9780387239460.

  29. Huser, R. and Davison, A. C. (2013) Composite Likelihood Estimation for the Brown–Resnick Process. Biometrika 100(2), 511–518.

  30. ——– (2014) Space-Time Modelling of Extreme Events. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76(2), 439–461.

  31. Huser, R., Davison, A. C. and Genton, M. G. (2016) Likelihood Estimators for Multivariate Extremes. Extremes 19(1), 79–103.

  32. Hüsler, J. and Reiss, R.-D. (1989) Maxima of Normal Random Vectors: Between Independence and Complete Dependence. Statistics & Probability Letters 7(4), 283–286.

  33. Jeon, S. and Smith, R. L. (2012) Dependence Structure of Spatial Extremes Using Threshold Approach. arXiv:1209.6344v1.

  34. Jun, M. and Stein, M. L. (2007) An Approach to Producing Space-Time Covariance Functions on Spheres. Technometrics 49(4), 468–479.

  35. ——– (2008) Nonstationary Covariance Models for Global Data. Annals of Applied Statistics 2(4), 1271–1289.

  36. Kabluchko, Z. and Schlather, M. (2010) Ergodic Properties of Max-Infinitely Divisible Processes. Stochastic Processes and their Applications 120(3), 281–295.

  37. 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.

  38. Lantuéjoul, C., Bacro, J.-N. and Bel, L. (2011) Storm Processes and Stochastic Geometry. Extremes 14(4), 413–428.

  39. Lindgren, F., Rue, H. and Lindström, J. (2011) An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73(4), 423–498.

  40. Lindsay, B. G. (1988) Composite Likelihood Methods. Contemporary Mathematics 80, 221–239.

  41. Marcon, G., Padoan, S. A., Naveau, P. and Muliere, P. (2014) Multivariate Nonparametric Estimation of the Pickands Dependence Function Using Bernstein Polynomials. arXiv:1405.5228v2.

  42. Ng, C. T. and Joe, H. (2014) Model Comparison with Composite Likelihood Information Criteria. Bernoulli 20(4), 1738–1764.

  43. Nikoloulopoulos, A. K., Joe, H. and Li, H. (2009) Extreme Value Properties of Multivariate \(t\) Copulas. Extremes 12(2), 129–148.

  44. Northrop, P. J. and Jonathan, P. (2011) Threshold Modelling of Spatially-Dependent Non-Stationary Extremes with Application to Hurricane-Induced Wave Heights (with Discussion). Environmetrics 22, 799–809.

  45. Nychka, D., Wikle, C. K. and Royle, J. A. (2002) Multiresolution Models for Nonstationary Spatial Covariance Functions. Statistical Modelling 2(4), 315–331.

  46. Opitz, T. (2013) Extremal \(t\) Processes: Elliptical Domain of Attraction and a Spectral Representation. Journal of Multivariate Analysis 122(1), 409–413.

  47. Paciorek, C. J. and Schervish, M. (2006) Spatial Modelling Using a New Class of Nonstationary Covariance Functions. Environmetrics 17(5), 483–506.

  48. 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.

  49. Perrin, O. and Monestiez, P. (1999) Modelling of Non-Stationary Spatial Structure Using Parametric Radial Basis Deformations. In geoENV II – Geostatistics for Environmental Applications, eds J. Gómez-Hernández, A. Soares and R. Froidevaux, volume 10 of Quantitative Geology and Geostatistics, pp. 175–186. Springer.

  50. Reich, B. J., Eidsvik, J., Guindani, M., Nail, A. J. and Schmidt, A. M. (2011) A Class of Covariate-Dependent Spatiotemporal Covariance Functions For The Analysis of Daily Ozone Concentration. Annals of Applied Statistics 5(4), 2465–2487.

  51. Reich, B. J. and Shaby, B. A. (2012) A Hierarchical Max-Stable Spatial Model for Extreme Precipitation. Annals of Applied Statistics 6(4), 1430–1451.

  52. Ribatet, M. (2013) Spatial Extremes: Max-Stable Processes at Work. Journal de la Société Française de Statistique 154(2), 156–177.

  53. Ribatet, M., Cooley, D. S. and Davison, A. C. (2012) Bayesian Inference from Composite Likelihoods, with an Application to Spatial Extremes. Statistica Sinica 22(2), 813–845.

  54. Sampson, P. D. and Guttorp, P. (1992) Nonparametric Estimation of Nonstationary Spatial Covariance Structure. Journal of the American Statistical Association 87(417), 108–119.

  55. Schlather, M. (2002) Models for Stationary Max-Stable Random Fields. Extremes 5(1), 33–44.

  56. Schmidt, A. M. and O’Hagan, A. (2003) Bayesian Inference for Non-Stationary Spatial Covariance Structure via Spatial Deformations. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65(3), 743–758.

  57. Smith, E. L. and Stephenson, A. G. (2009) An Extended Gaussian Max-Stable Process Model for Spatial Extremes. Journal of Statistical Planning and Inference 139(4), 1266–1275.

  58. Smith, R. L. (1990) Max-Stable Processes and Spatial Extremes. Unpublished.

  59. Stein, M. L. (1999) Interpolation of Spatial Data: Some Theory for Kriging. First edition. New York: Springer. ISBN 9780387986296.

  60. ——– (2005) Nonstationary Spatial Covariance Functions. Unpublished.

  61. Stephenson, A. and Tawn, J. A. (2005) Exploiting Occurrence Times in Likelihood Inference for Componentwise Maxima. Biometrika 92(1), 213–227.

  62. Thibaud, E., Aalto, J., Cooley, D. S., Davison, A. C. and Heikkinen, J. (2015) Bayesian Inference for the Brown–Resnick Process, With an Application to Extreme Low Temperatures. arXiv:1506.07836v1.

  63. Thibaud, E., Mutzner, R. and Davison, A. C. (2013) Threshold Modeling of Extreme Spatial Rainfall. Water Resources Research 49(8), 4633–4644.

  64. Thibaud, E. and Opitz, T. (2015) Efficient Inference and Simulation for Elliptical Pareto Processes. Biometrika 102(4), 855–870.

  65. Varin, C., Reid, N. and Firth, D. (2011) An Overview of Composite Likelihood Methods. Statistica Sinica 21, 5–42.

  66. Wadsworth, J. L. and Tawn, J. A. (2012) Dependence Modelling for Spatial Extremes. Biometrika 99(2), 253–272.

  67. ——– (2014) Efficient Inference for Spatial Extreme Value Processes Associated to Log-Gaussian Random Functions. Biometrika 101(1), 1–15.

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Huser, R., Genton, M.G. Non-Stationary Dependence Structures for Spatial Extremes. JABES 21, 470–491 (2016).

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  • Covariate
  • Extremal t model
  • Extreme event
  • Max-stable process
  • Non-stationarity