Spatial Modelling of Annual Max Temperatures using Max Stable

Processes

NCAR Advanced Study Program24 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/i

ndex.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)– 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

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:– North Atlantic Oscillation (NAO)

Positive Phase

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– (an)isotropic covariance (Smith)– Whittle-Matérn, Stable, Powered Exponential,

Cauchy

*Ribatet ASP .ppt (2011)

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,elevScale: lon, avg temp

Shape: lat, lon, lat*lon

Location: lat, lon,elev,NAOScale: 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

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

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 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.

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)........