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S p atial Modelling of Annual Max Temperatures using Max Stable Processes

S p atial Modelling of Annual Max Temperatures using Max Stable Processes. NCAR Advanced Study Program 24 June, 2011 Anne Schindler, Brook Russell, Scott Sellars, Pat Sessford , and Daniel Wright. Outline. Introduction to spatial extreme value analysis R package— SpatialExtremes

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S p atial Modelling of Annual Max Temperatures using Max Stable Processes

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  1. Spatial Modelling of Annual Max Temperatures using Max Stable Processes NCAR Advanced Study Program 24 June, 2011 Anne Schindler, Brook Russell, Scott Sellars, Pat Sessford, and Daniel Wright

  2. Outline • Introduction to spatial extreme value analysis • R package—SpatialExtremes • Study area and data • Covariates • Modeling fitting and results • Summary • Challenges and future work

  3. Introduction to Spatial Extremes • Societal impacts of extreme events • Extreme value analysis of physical processes • Temperature • Precipitation • Streamflow • Waves • Characterization of the spatial dependency of extreme events

  4. R package—SpatialExtremes • Developed by Dr. Mathieu Ribatet • http://spatialextremes.r-forge.r-project.org/index.php • Several techniques for analyzing spatial extremes: • Gaussian copulas • Bayesian hierarchical model (BHM) • Max stable processes • Simulation

  5. Study Area and Data • DWD Met Stations (36) • State of Hessen, Germany • Annual Max Temperature (Apr-Sept) • Elevation from 110 to 921 meters • Maximum separation distance of 200 km • Modeling data set • 16 Stations (1964-2006) with 24 years of overlapping data • Cross-validation data set • 8 stations with 10 years overlapping • 5 stations with 40 years overlapping Germany Wikipedia.com

  6. Station Locations Germany Wikipedia.com • State of Hessen, Germany

  7. Covariates • Spatial Covariates: • Latitude and Longitude • Magnitude of extreme events might be different depending on location • Elevation • Avg. Summer Temp

  8. Covariates Positive Phase • Temporal Covariate: • North Atlantic Oscillation (NAO) Negative Phase http://www.ldeo.columbia.edu/res/pi/NAO/

  9. Modeling Framework • No Blue Print to follow! • Fit Marginal GEVs (station by station) • Estimate spatial dependence • Pick model for max stable process • Pick correlation structure • Estimate marginals • Select covariates for trend surfaces • Fit max stable model using pairwise likelihood

  10. Models For Max Stable Process • Candidate models • Correlation Structure • (an)isotropic covariance (Smith) • Whittle-Matérn, Stable, Powered Exponential, Cauchy *Ribatet ASP .ppt (2011)

  11. Model Fitting Criteria • TIC • Madogram • Parameter estimates (station by station vs. spatial marginals)

  12. Station By Station (GEV)

  13. Spatial Dependence (Madogram)

  14. Spatial Dependence (Madogram)

  15. Geometric-Gaussian Model: Different Covariates Location: lat, lon,elev Scale: lon, avg temp Shape: lat, lon, lat*lon Location: lat, lon,elev,NAO Scale: lon Shape: lat, lon, lat*lon

  16. Parameter Estimates

  17. Estimated Return Levels

  18. Summary • High spatial dependence in annual maximum temperature in research area (Hessen) • Spatial covariates for shape parameter fairly complexno literature to support this (only precip examples ) • Most models and covariate combinations underestimated the spatial dependence of the data • Different optimization methods gave different results

  19. Challenges and Future Work • New field of EVA, lack of examples • Spatial dependence greatly varies with earth science variables (temperature vs. precipitation) • Small regions vs. large regions (dependence structure?) • Computational issues? • Optimization/composite likelihood issues • Uncertainty estimation • Simulations • Applications?

  20. Extra Bonus Quiz: Who Said It? • “If you can’t solve the problem, change the problem.” • “If you want to stay awake, do not go into that talk!” • “Loading…”

  21. Thank you!Questions and Comments?

  22. References • de Haan, L. (1984). A spectral representation for max-stable processes. The Annals of Probability, 12(4):1194-1204. • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. • Cooley, D., Naveau, P., and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Springer, editor, Dependence in Probability and Statistics, volume 187, pages 373-390. Springer, New York, lecture notes in statistics edition. • Kabluchko, Z., Schlather, M., and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Prob., 37(5):2042-2065. • Lindsay, B. (1988). Composite likelihood methods. Statistical Inference from Stochastic Processes. American Mathematical Society, Providence. • Padoan, S., Ribatet, M., and Sisson, S. (2010). Likelihood-based inference for max-stable processes. Journal of the American Statistical Association (Theory & Methods), 105(489):263-277. • Schlather, M. (2002). Models for stationary max-stable random fields. Extremes, 5(1):33-44. • Smith, R. L. (1990). Max-stable processes and spatial extreme. Unpublished manuscript.

  23. ENSEMBLES Project • RCMs covering Europe, driven by GCMs or reanalysis data (1958-2002). • Here we focus on the Hessen (a state in Deutschland) area, with the data driven by reanalysis.....

  24. Observations vs. Climate Model • Location parameters differ in places but agree on a lot, but the scale and shape parameters disagree completely; presumably the observational data are more realistic. • BUT.... possible inconsistencies when extrapolating out of the spatial range of observation stations?? (Whereas this is not an issue with data from climate models)........

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