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New approaches for extreme value analysis in large-scale geospatial-temporal data with applications to observed and climate-model simulated precipitation in South America
Gabriel Kuhn, Shiraj Khan, Auroop R Ganguly*
Oak Ridge National Laboratory, Oak Ridge, TN
* Presenter / Correspondence: [email protected]; 865-241-1305
2006 Fall Meeting of the American Geophysical Union, San Francisco, CA
Section: Hydrology
Session: Role of Observed Precipitation in Atmospheric and Land Surface Models I
Paper #: H32A-07
13 December, 2006
The following article has been submitted to a journal after this specific abstract was submitted to AGU 2006Kuhn, G., Khan, S., Ganguly, A.R.*, and M. Branstetter (2006): Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources (In Review). * Corresponding Author
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Motivation
Extreme Value Theory• Prob. (X > u): Univariate Extreme Value
Theory (EVT)
• Prob. (Y > v | X = x): Extremes and Multiple Covariates
• Prob. (Y > v | X > u): Multivariate Extreme Value Theory
• Hydrologic Observations
− Grid-based Precipitation
• Climate Model Simulations
− 1870 to Now: Past
− Now to 2100: Future
1870 Now 2100
Observations
Model Simulations
Gaps for real applications• Large-scale data: Scale up extreme
value theory for automated use
• Geospatial-temporal extremes: Extend multivariate EVT for space and time
Quantify Model Uncertainties
Generate Realistic Prediction Scenarios
Khan, S., Kuhn, G., Ganguly, A.R.*, Erickson, D.J., and G. Ostrouchov (2006): Spatio-temporal variability of daily and weekly Precipitation extremes in South America, Water Resources Research (In Review).
Kuhn, G., Khan, S., Ganguly, A.R.*, and M. Branstetter (2006): Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources (In Review).
Kuhn. G., Khan, S., and Ganguly, A.R.* (2006): New approaches for extreme value analysis in large-scale geospatial-temporal data with applications to observations and climate-model simulated precipitation in South America. American Geophysical Union, Fall Meeting, SFO, CA.
* Corresponding Author
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Precipitation Data in South America
• Historical Precipitation− Grid-based: 1 degree spatial grids
− Daily data from January 1940 to June 2005
− NOAA data: Liebman and Allured, 2005, BAMS
• Simulated Precipitation − T85 grid: 1.4 degree over land and atmosphere
− Daily and 6-hourly data from 1940-2099
− IPCC runs from Community Climate System Model version 3 (CCSM3): Collins et al., 2005
− “A2” IPCC Scenario (PCMDI at LLNL)
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Extreme Value Theory: One VariableGeneralized Pareto Distribution & Return Level
EXCEEDENCES OVER THRESHOLD: Prob. (X – u | X > u)
• T-year Return Level, RL(T)− Exceeded once every T years
− Prob. [X > RL(T)] in any year: 1/T
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Population Threat
Metrics & Uncertainty
Precipitation
Extremes
Geo-Referenced Indices
for Disaster Readiness
Overview
South
America
Grid-based Precipitation
(Daily; 1o, 2.5o grids)
Sabesan, A., Abercrombie, K., Ganguly, A.R.*, Bhaduri, B.L., Bright, E.A., and P. Coleman (2006): Metrics for the comparative analysis of geospatial datasets with applications to high-resolution grid-based population data, GeoJournal (Invited: In Review).
Khan, S., Kuhn, G., Ganguly, A.R.*, Erickson, D.J., and G. Ostrouchov (2006): Spatio-temporal variability of daily and weekly Precipitation extremes in South America, Water Resources Research (In Review).
* Corresponding AuthorFuller, C.T., Sabesan, A., Khan, S., Kuhn. G., Ganguly, A.R.*, Erickson,
D., and G. Ostrouchov (2006): Quantification and visualization of the human impacts of anticipated precipitation extremes in South America. American Geophysical Union, Fall Meeting, San Francisco, CA.
Global Extent
LandScanTM Global 2004GIST Group, CSE Division, ORNL
Grid-based Population Database
(30// lat-lon)
Precipitation Grids (2005) ESRL, PS Division, NOAA
Spatial Statistics / GIS Extreme Value Theory
Geospatial Modeling
Gross Domestic ProductCIA World Factbook 2006
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Geospatial Correlation to Geospatial-Temporal Dependence
• Spatial Correlation Functions (Cressie, 1993)− Captures linear correlation and works well for multivariate normal
− Spatial extensions of ACF and CCF used in time series analysis
− Spatial ACF and spatial CCF are function of spatial lags
• Kendall’s Tau (Kendall and Gibbons, 1990)− Captures linear correlation and monotonic dependence
− Function at spatial lags analogous to spatial correlation function
• Spatio-Temporal Correlation and Dependence− Relates time series at multiple spatial grids or points
− Linear: Cross-correlation among time series at multiple spatial locations
− Linear + Monotonic dependence: Kendall’s Tau for the above
• Measures for Complete Dependence Structures− Information theoretic (Mutual Information): Khan et al., 2006, GRL
− Copulas: Described in later slides
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
The Kendall Tau for Dependence
Kuhn, G., Khan, S., Ganguly, A.R.*, and M. Branstetter (2006): Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources (In Review).
Kendall’s Tau: Definition
Empirical Estimator for iid Samples
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceMotivation
• Conditional Exceedence
− Prob. (X > u | Y > v)
− Extremes of river flows conditional on extremes of El Nino?
• Joint Exceedence
− Prob. (X > u, Y > v)
− Precipitation extremes of nearby spatial locations co-occur?
• Joint and conditional probabilities are related
• Applications
− Extreme dependence among high-risk variables?
− Are there regions where heat waves may co-occur with significant storms?
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceThe Concept of Copula
• Measure of dependence structure
− Joint distribution: Marginal distribution AND dependence
− Copula: Quantifies dependence from joint distributions by combining univariate distributions in a specific way
Figures courtesy Dorey & Joubert (2005):Dorey, M., and P. Joubert (2005): Modeling Copulas: An Overview, The Staple Inn Actuarial Society, 27 pages.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceCopula Definition & Sklar’s Theorem
• “Multivariate CDF defined on the n-dimensional unit cube [0, 1]n such that every marginal distribution is uniform on the interval [0, 1]”− Complete information on variable dependence
− No information on marginal distributions
• C(u,0) = 0 = C(0,v); C(u,1) = u; C(1,v) = v
• Sklar’s Theorem (Bivariate): H(x,y) = C(F(x),G(y))− H(x,y): Bivariate Distribution
− F(x), G(y): Marginal DistributionsG(y)=H((-∞, ∞),y); F(x)=H(x,(-∞, ∞));
− F(x) & G(y) continuous � C is uniqueIf not, C is unique on the range of values of the marginals
Courtesy: “Wikipedia, the free encyclopedia”
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceTail Copula and Tail Dependence
Definition of the Tail Copula of XXXX
Pair-wise Tail Dependence Coefficients
Empirical Estimation for Elliptical CopulaKuhn, G., Khan, S., Ganguly, A.R.*, and M. Branstetter (2006): Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources (In Review).
Kuhn, G. (2006): On dependence and extremes, Ph.D. thesis, Munich University of Technology. (Chapter 3).
Pair-wise Geospatial-Temporal Tail Dependence
Time series at two pairs of spatial locations (grids)
Pair-wise tail dependence based on tail dependence measure
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceIntuition on Tail Dependence
Kuhn, G., Khan, S., Ganguly, A.R.*, and M. Branstetter (2006): Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources (In Review).
• Location X has t-year return level, RL(t), of zx
• Location Y has t-year return level of zy
• Locations X and Y simultaneously exceed RL(t)
Simultaneous exceedence of RL(t) is a (t/λλλλxy)-year event
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Multivariate Tail DependenceIntuition on Tail Dependence
• 100-year precipitation for location X is RL(100;X)
• 100-year precipitation for location Y is RL(100;Y)
• Prob. (X > RLX(100)) = 1/100
• Prob. (Y > RLY(100)) = 1/100
• Case A: Processes at locations X & Y are independent − Prob. ( (X > RLX(100)), (Y > RLY(100))) = (1/100)*(1/100)
− Simultaneous exceedence of 100-year level is a 10000-year event!!
• Case B: Processes at locations X & Y are dependent − Consider a λXY of 0.5 and t of 100
− Prob. ( (X > RLX(t)), (Y > RLY(t))) ~= λXY / t = 1/200
− Simultaneous exceedence is a mere 200-year event!!
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Geospatial-Temporal Extreme DependenceLumped Temporal & Spatio-Temporal Dependence
Observations
Simulations
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Geospatial-Temporal Extreme DependencePair-wise dependence: One selected point (X) with all others
Observations Simulations
Correlation on left columns
Tail Dependence on right columns
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Geospatial-Temporal Extreme DependenceTrends in Pair-wise Dependence
Observations Simulations
Correlation on bottom row; Tail dependence in top row
Results using (1965-1990) data on the left and (1980-2005) on the right
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Insights and Comparisons
• Average and pair-wise dependence* in simulations and observations show good match on the whole
• Differences between simulated and observed dependence*
− Average dependence* is higher for simulated data
− Average dependence* within observations show increasing trend but simulations exhibit no such trend
− Average simulated dependence* exhibit a longitudinal tilt but observed dependence* do not
− Pair-wise dependence* is higher and more spread-out in simulations compared to observations
− Pair-wise dependence* exhibits subtle differences in specific instances, both in space and in time
* Dependence ���� Correlation and tail dependence
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Future Research
• Statistical Methodologies: Important Gaps and Issues
• Multiple Regions: Continental US, Sahel region of Africa, etc.
• Multiple Variables: Heat Waves and Precipitation Extremes
• Long-range Extreme Dependence: El Nino and Precipitation
• Model Evaluation: Uncertainty & Degree of Belief
• Predictive Scenarios: Scenario-based Predictions
• Feedback: Model Improvements
• Decision-making: Visualization and collaboration tools
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Backup Slides“Copula for Geophysicists”
• This preliminary “tutorial” on copula uses the following sources
− Our manuscript (Kuhn et al., 2006)
− Clemen, R.T., and T, Reilly (1999): Correlations and Copula for Decisions and Risk Analysis, Management Science, 45(2): 208-224.
− Li, D. X. (2000): On Default Correlation: A Copula Function Approach, The RiskMetrics Group, Working Paper Number 99-07.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Copula Tutorial – (1)
From Li (2000)
Note: This is a tutorial and not presentation of original results /
write-ups and/or results / write-ups
generated by any of the authors.
The reproduction is almost an exact copy of the reference cited.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Copula Tutorial – (2)
From Clemen and
Reilly (1999)Note: This is a tutorial and not
presentation of original results /
write-ups and/or results / write-ups
generated by any of the authors. The reproduction is almost an
exact copy of the reference cited.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Copula Tutorial – (3)
From Li (2000)
Note: This is a tutorial and not presentation of original results /
write-ups and/or results / write-ups
generated by any of the authors.
The reproduction is almost an exact copy of the reference cited.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Copula Tutorial – (3)
From Li (2000)
Note: This is a tutorial and not presentation of original results /
write-ups and/or results / write-ups
generated by any of the authors.
The reproduction is almost an exact copy of the reference cited.
OAK RIDGE NATIONAL LABORATORY
U. S. DEPARTMENT OF ENERGY
Acknowledgments and Copyrights
This research was sponsored by the SEED money funds of the Laboratory Directed Research and Development program of the Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC for the U. S. Department of Energy under contract no. DEAC05-00OR22725. (Title of SEED project: Multivariate dependence in climate extremes; Principal Investigator: Auroop R. Ganguly).
Auroop Ganguly would like to thank Professor Tailen Hsing of Ohio State, Dr. Rick Katz of NCAR, as well as Drs. David J Erickson III, George Ostrouchov and Marcia Branstetter of ORNL for supporting and participating in the SEED project, Drs. Budhendra L. Bhaduri, John B. Drake, and Virginia H. Dale of ORNL for their helpful comments, and Professor Sunil Saigal of the University of South Florida for his help. The reviews from all ORNL-internal and external publications or manuscripts that contributed to this research are all gratefully acknowledged.
This work was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DEAC05-00OR22725. This work has been authored by employees and contractors of the U.S. Government, accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S Government purposes.