Extreme Values on Spatial Fields p. 1/1

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1 Extreme Values on Spatial Fields Daniel Cooley Department of Applied Mathematics, University of Colorado at Boulder Geophysical Statistics Project, National Center for Atmospheric Research Philippe Naveau Department of Applied Mathematics, University of Colorado at Boulder Laboratiore des Sciences du Climat et de l Environnement, IPSL-CNRS, Gif-sur-Yvette, France Doug Nychka Geophysical Statistics Project, National Center for Atmospheric Research Extreme Values on Spatial Fields p. 1/1

2 Background 5th year graduate student 2nd year with GSP Anticipated graduation date: Fall 2005 Extreme Values on Spatial Fields p. 2/1

3 Background 5th year graduate student 2nd year with GSP Anticipated graduation date: Fall 2005 Projects Extreme value model for Lichenometry Spatial dependence estimation and prediction for annual maxima Colorado Front Range Precipitation Extreme Values on Spatial Fields p. 2/1

4 Colorado Precipitation Project Goal: Create a map of precipitation return levels for Colorado s Front Range. Extreme Values on Spatial Fields p. 3/1

5 Colorado Precipitation Project Goal: Create a map of precipitation return levels for Colorado s Front Range. project originated from ISSE, interested in flooding NWS has done maps for Southwest US (AZ, NM, UT, NV) and mid-atlantic (OH, PA, MD, NJ, NC, WV) plans to do entire US (contingent on funding) handles spatial dependence and prediction differently present NWS with our method and results Extreme Values on Spatial Fields p. 4/1

6 Front Range Data Data: hourly precipitation from 56 weather stations, years of data, Apr1 - Oct 31. latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo Limon Extreme Values on Spatial Fields p. 5/1

7 Univariate Extreme Values GEV: Used to model block (annual) maxima wasteful of data used by NWS to produce their maps Extreme Values on Spatial Fields p. 6/1

8 %& $ "! # ' Univariate Extreme Values GEV: Used to model block (annual) maxima wasteful of data used by NWS to produce their maps GPD: Models exceedences above a threshold # where Less wasteful of data Must choose threshold How to go spatial? Extreme Values on Spatial Fields p. 6/1

9 /. 3 Bayesian Hierarchical Model Let - (*),+ be observation from station. : 9 ) ) (*) + Extreme Values on Spatial Fields p. 7/1

10 /. 3 : 0 : Bayesian Hierarchical Model Let - (*),+ be observation from station. : 9 ) ) (*) + 4BA > ;=< 4BD 8 D A 8 C 8 are functions of covariates are functions of distance Extreme Values on Spatial Fields p. 7/1

11 /. 3 : 0 : Bayesian Hierarchical Model Let - (*),+ be observation from station. : 9 ) ) (*) + 4BA > ;=< 4BD 8 D A 8 C 8 are functions of covariates are functions of distance Plan: Use existing spatial techniques on the parameters, then convert to the desired return levels. Extreme Values on Spatial Fields p. 7/1

12 Covariates To do spatial prediction, any covariate must be available for every location in the region. sigma xi log(sigma) xi elevation elevation Extreme Values on Spatial Fields p. 8/1

13 PO # N Q PO I 4 c: : 4 C A [ [ 4 4 c: : 3 D [ [ : : k k First Model E E # J I H BEGF S Q R KML I J T distance \ W `ba _ ]^ Y \ W U elevation V W Z Y VXW U distance \ d `ba _ ]^ Y \ d U elevation Vd Z Y Vd U k 8 4 j a / g h 0 \ ee k 8 4 j a / gih 0 Vfee Extreme Values on Spatial Fields p. 9/1

14 Model Results log(sigma) xi elevation elevation Extreme Values on Spatial Fields p. 10/1

15 s o Model Results prq lnm latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo longitude Limon latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo longitude Limon Extreme Values on Spatial Fields p. 11/1

16 Model Results latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo longitude Limon latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo longitude Limon Extreme Values on Spatial Fields p. 12/1

17 Model Results 20-year Return Levels latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo Limon latitude Breckenridge Ft. Collins Greeley Boulder Denver Colo Spgs Pueblo Limon longitude longitude Extreme Values on Spatial Fields p. 13/1

18 Future Work Extend spatially Search for covariates Speed up MCMC method Examine covariance structure Test other models Model comparison (DIC)??? Extreme Values on Spatial Fields p. 14/1