Spatial Extremes

Spatial Modelling of Annual Max
Temperatures using Max Stable
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
Modeling fitting and results
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
• Several techniques for analyzing spatial extremes:
– Gaussian copulas
– Bayesian hierarchical model (BHM)
– Max stable processes
– Simulation
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
Station Locations
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:
Positive Phase
– North Atlantic Oscillation (NAO)
Negative Phase
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
• Madogram
• Parameter estimates (station by station vs. spatial
Station By Station (GEV)
Spatial Dependence (Madogram)
Spatial Dependence (Madogram)
Geometric-Gaussian Model: Different
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
• High spatial dependence in annual maximum
temperature in research area (Hessen)
• Spatial covariates for shape parameter fairly
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
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
Extra Bonus Quiz: Who Said It?
a) “If you can’t solve the problem, change the
b) “If you want to stay awake, do not go into that
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.,
• 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
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

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