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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 Outline • • • • • • • Introduction to spatial extreme value analysis R package—SpatialExtremes Study area and data Covariates Modeling fitting and results Summary Challenges and future work 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 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 Study Area and Data • DWD Met Stations (36) – – – – Germany 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 Wikipedia.com Station Locations Germany Wikipedia.com State of Hessen, Germany Covariates • Spatial Covariates: – Latitude and Longitude • Magnitude of extreme events might be different depending on location – Elevation – Avg. Summer Temp Covariates • Temporal Covariate: Positive Phase – North Atlantic Oscillation (NAO) Negative Phase http://www.ldeo.columbia.edu/res/pi/NAO/ 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 Models For Max Stable Process • Candidate models • Correlation Structure *Ribatet ASP .ppt (2011) – (an)isotropic covariance (Smith) – Whittle-Matérn, Stable, Powered Exponential, Cauchy Model Fitting Criteria • TIC • Madogram • Parameter estimates (station by station vs. spatial marginals) Station By Station (GEV) Spatial Dependence (Madogram) Spatial Dependence (Madogram) 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 Parameter Estimates Estimated Return Levels Summary • High spatial dependence in annual maximum temperature in research area (Hessen) • Spatial covariates for shape parameter fairly complex no 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 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? Extra Bonus Quiz: Who Said It? a) “If you can’t solve the problem, change the problem.” b) “If you want to stay awake, do not go into that talk!” c) “Loading…” Thank you! Questions and Comments? 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 maxstable 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 maxstable 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. 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..... 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)........