1. NCAR Advanced Study Program
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
Several techniques for analyzing spatial extremes:
Gaussian copulas
Bayesian hierarchical model (BHM)
Max stable processes
Simulation

5. Study Area and Data Cross-validation data set 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 ( ) 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 Elevation
Magnitude of extreme events might be different depending on location
Elevation
Avg. Summer Temp

8. Covariates Temporal Covariate: North Atlantic Oscillation (NAO)
Positive Phase
Temporal Covariate:
North Atlantic Oscillation (NAO)
Negative Phase

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 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

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):
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 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):
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):
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 ( ).
Here we focus on the Hessen (a state in Deutschland) area, with the data driven by reanalysis.....

29. 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)