Gabriel Kuhn, Shiraj Khan, Auroop R Ganguly* Oak Ridge National Laboratory, Oak Ridge, TN

Transcription

1 13 December, 2006 New approaches for extreme value analysis in large-scale geospatialtemporal 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: gangulyar@ornl.gov; The following article has been submitted to a journal after this specific abstract was submitted to AGU 2006 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). * Corresponding Author 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

2 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 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 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). Hydrologic Observations Grid-based Precipitation Climate Model Simulations 1870 to Now: Past Now to 2100: Future Quantify Model Uncertainties Generate Realistic Prediction Scenarios Observations Model Simulations 1870 Now 2100 Khan, S., Kuhn, G., Ganguly, A.R.*, Erickson, D.J., and G. Ostrouchov (2006): Spatiotemporal variability of daily and weekly Precipitation extremes in South America, Water Resources Research (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

3 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 IPCC runs from Community Climate System Model version 3 (CCSM3): Collins et al., 2005 A2 IPCC Scenario (PCMDI at LLNL)

4 Extreme Value Theory: One Variable Generalized 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

5 Overview Grid-based Population Database (30 // lat-lon) Global Extent South America Grid-based Precipitation (Daily; 1 o, 2.5 o grids) LandScan TM Global 2004 GIST Group, CSE Division, ORNL Spatial Statistics / GIS Extreme Value Theory Precipitation Grids (2005) ESRL, PS Division, NOAA Population Threat Metrics & Uncertainty Geospatial Modeling Precipitation Extremes 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). Gross Domestic Product CIA World Factbook 2006 Geo-Referenced Indices for Disaster Readiness Fuller, 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. Khan, S., Kuhn, G., Ganguly, A.R.*, Erickson, D.J., and G. Ostrouchov (2006): Spatiotemporal variability of daily and weekly Precipitation extremes in South America, Water Resources Research (In Review). * Corresponding Author

6 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

7 The Kendall Tau for Dependence Kendall s Tau: Definition Empirical Estimator for iid Samples 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).

8 Multivariate Tail Dependence Motivation 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?

9 Multivariate Tail Dependence The 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.

10 Multivariate Tail Dependence Copula 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 Distributions G(y)=H((-, ),y); F(x)=H(x,(-, )); F(x) & G(y) continuous C is unique If not, C is unique on the range of values of the marginals Courtesy: Wikipedia, the free encyclopedia

11 Multivariate Tail Dependence Tail Copula and Tail Dependence Definition of the Tail Copula of X Pair-wise Tail Dependence Coefficients Pair-wise Geospatial-Temporal Tail Dependence Time series at two pairs of spatial locations (grids) Pair-wise tail dependence based on tail dependence measure Empirical Estimation for Elliptical Copula 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. (2006): On dependence and extremes, Ph.D. thesis, Munich University of Technology. (Chapter 3).

12 Multivariate Tail Dependence Intuition on Tail Dependence Location X has t-year return level, RL(t), of z x Location Y has t-year return level of z y Locations X and Y simultaneously exceed RL(t) Simultaneous exceedence of RL(t) is a (t/λ xy )-year event 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).

13 Multivariate Tail Dependence Intuition 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 > RL X (100)) = 1/100 Prob. (Y > RL Y (100)) = 1/100 Case A: Processes at locations X & Y are independent Prob. ( (X > RL X (100)), (Y > RL Y (100))) = (1/100)*(1/100) Simultaneous exceedence of 100-year level is a year event!! Case B: Processes at locations X & Y are dependent Consider a λ XY of 0.5 and t of 100 Prob. ( (X > RL X (t)), (Y > RL Y (t))) ~= λ XY / t = 1/200 Simultaneous exceedence is a mere 200-year event!!

14 Geospatial-Temporal Extreme Dependence Lumped Temporal & Spatio-Temporal Dependence Observations Simulations

15 Geospatial-Temporal Extreme Dependence Pair-wise dependence: One selected point (X) with all others Correlation on left columns Tail Dependence on right columns Observations Simulations

16 Geospatial-Temporal Extreme Dependence Trends in Pair-wise Dependence Observations Simulations Correlation on bottom row; Tail dependence in top row Results using ( ) data on the left and ( ) on the right

17 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

18 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

19 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): Li, D. X. (2000): On Default Correlation: A Copula Function Approach, The RiskMetrics Group, Working Paper Number

20 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.

21 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.

22 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.

23 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.

24 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. DEAC05-00OR (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. DEAC05-00OR 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.