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

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

• Societal impacts of extreme events
• Extreme value analysis of physical processes
– Temperature
– Precipitation
– Streamflow
– Waves
• Characterization of the spatial dependency of extreme events

• 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

Germany
• 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
Wikipedia.com

Germany
Wikipedia.com
State of Hessen, Germany

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

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

• 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

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

• TIC
• Madogram
• Parameter estimates (station by station vs. spatial marginals)

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

• 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

• 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?

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?

• 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.

• 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.....

• 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)........