Generating gridded fields of extreme precipitation for large domains with a Bayesian hierarchical model

Transcription

1 Generating gridded fields of extreme precipitation for large domains with a Bayesian hierarchical model Cameron Bracken Department of Civil, Environmental and Architectural Engineering University of Colorado at Boulder Vannes, France May 19, / 24

2 Coauthors Balaji Rajagopalan - University of Colorado Will Kleiber - University of Colorado Linyin Cheng - NOAA Subhrendu Gangopadhyay - Bureau of Reclamation 2 / 24

3 This talk Motivation - Link to weather generation Bayesian hierarchical model (BHM) for spatial extremes Application to the Western US Simuion procedure using the BHM 3 / 24

4 Introduction Background BHM Application Simuion References Motivation - Extremes in the western US? I From the Western States (WA, OR, CA, ID, NV, UT, AZ, MT, WY, CO, NM, ND, SD, NE, KS, OK, and TX) experienced $24.7 billion in flood damages, $1.5 billion annually. I California, Washington, and Oregon alone accounted for $10.6 billion (46 percent) [Ralph et al., 2014, Pielke et al., 2002] Boulder Flood, 2013 Research into hydroclimate extremes can provide more accurate and localized estimates of flood risk. 4 / 24

5 More Motivation How are simued fields of extremes linked to weather generation? May not need a full weather generator extremes may be the only quantity of interest Preserve spatial dependence of simued extremes Downscaling 5 / 24

6 What are BHM Spatial Extremes models typically used for? Return Levels 100 year return level [cm] year return level 90% CI [cm] lon lon 6 / 24

7 Background on Bayesian Statistics From Bayes rule: θ Parameters y Dependent data (response) p(θ y, x) p(y θ, x) p(θ, x) }{{}}{{}}{{} Posterior Likelihood Prior x Independent data (covariates/predictors/constants) Posterior: The answer, probability distributions of parameters Likelihood: A computable function of the parameters, model specific Prior: Probability distribution, incorporates existing knowledge of the system, model specific 7 / 24

8 Bayesian Hierarchical Model In a hierarchical Bayesian model, expand terms using conditional distributions where θ = (θ 1, θ 2 ): p(θ y) p(y θ 1 ) }{{}}{{} Posterior p(θ 1 θ 2 ) }{{} Data Likelihood Process Liklihood p(θ 2 ) }{{} Prior Data Likelihood Rees observed data to distribution parameters Process Likelihood Rees distribution parameters of the to each other 8 / 24

9 Statistics of Extremes Given daily data, if we select the maximum value in each year, those data follow a generalized extreme value (GEV) distribution: b = 1 + ξ GEV(y; µ, σ, ξ) = 1 σ b( 1/ξ) 1 exp { b 1/ξ} ( ) x µ σ, µ: Location, σ: Scale, ξ: Shape. Return Level (quantile function): 0.4 GEV(0,1,-0.5) GEV(0,1,0) GEV(0,1,0.5) z r = µ + σ ξ [( log(1 1/r)) ξ 1] Where r is the return period in years (100 years for example). f(y) y 9 / 24

10 Hierarchical spatial model (Y(s 1, t),..., Y(s n, t)) C g [Σ; {µ(s), σ(s), ξ(s)}] Y(s, t) GEV[µ(s), σ(s), ξ(s)] 10 / 24

11 Hierarchical spatial model (Y(s 1, t),..., Y(s n, t)) C g [Σ; {µ(s), σ(s), ξ(s)}] Y(s, t) GEV[µ(s), σ(s), ξ(s)] µ(s) = β µ,0 + xµ(s)β T µ (s) + w µ (s) σ(s) = β σ,0 + xσ(s)β T σ (s) + w σ (s) ξ(s) = β ξ,0 + xξ T(s)β ξ (s) + w ξ(s) 10 / 24

12 Hierarchical spatial model (Y(s 1, t),..., Y(s n, t)) C g [Σ; {µ(s), σ(s), ξ(s)}] Y(s, t) GEV[µ(s), σ(s), ξ(s)] µ(s) = β µ,0 + xµ(s)β T µ (s) + w µ (s) σ(s) = β σ,0 + xσ(s)β T σ (s) + w σ (s) ξ(s) = β ξ,0 + xξ T(s)β ξ (s) + w ξ(s) w µ (s) GP(0, C(θ µ )) w σ (s) GP(0, C(θ σ )) w ξ (s) GP(0, C(θ ξ )) 10 / 24

13 Hierarchical spatial model (Y(s 1, t),..., Y(s n, t)) C g [Σ; {µ(s), σ(s), ξ(s)}] Y(s, t) GEV[µ(s), σ(s), ξ(s)] µ(s) = β µ,0 + xµ(s)β T µ (s) + w µ (s) σ(s) = β σ,0 + xσ(s)β T σ (s) + w σ (s) ξ(s) = β ξ,0 + xξ T(s)β ξ (s) + w ξ(s) w µ (s) GP(0, C(θ µ )) w σ (s) GP(0, C(θ σ )) w ξ (s) GP(0, C(θ ξ )) Covariates: x T (s) = (Elevation, Mean Seasonal Precip) T C: Stationary, isotropic, exponential covariance model 10 / 24

14 Spatially varying regression coefficients β µ (s) = β σ (s) = β ξ (s) = k c µ i ηµ i (s) i=1 k ci σ ησ i (s) i=1 k c ξ i ηξ i (s) i=1 η i (s) = exp(d i /φ i ) 11 / 24

15 Elliptical copula for data dependence The Gaussian elliptical copula constructs the joint cdf of of a random vector (V 1,..., V n ) as F C (v 1,..., v n ) = Φ Σ (u 1,..., u n ) (1) where Φ Σ ( ) is the joint cdf of an n-dimensional multivariate normal distribution with covariance matrix Σ, u i = φ 1 (F i [v i ]), φ 1 is the inverse cdf (quantile function) of the standard normal distribution and F i ( ) is the marginal GEV cdf of variable i. V V 1 F2(V2) Cumued probability F 1(V 1) Normalized X Gaussian transformation Normalized X 1 Normalized X Multivariate Gaussian model Normalized X 1 12 / 24

16 Elliptical copula for data dependence The corresponding joint pdf is f C (y 1,..., y m ) = m i=1 m i=1 f i [y i ] Ψ Σ (u 1,...u m ) (2) ψ[u i ] where f i is the marginal GEV pdf at site i, ψ is the standard normal pdf and Ψ Σ is the joint pdf of an m-dimensional multivariate normal distribution. Exponential dependence matrix for spatial data: Σ(i, j) = exp( s i s j /a 0 ) (3) a 0 is the copula range parameter. Termed the dependogram since values are not covariances [Renard, 2011]. 13 / 24

17 Pros and Cons of Gaussian elliptical copulas Benefits: Easy to implement Few parameters Flexible (can use any marginal distribution) Requires checking two conditions in application: Asymptotic independence (dependence is hard to find in practice) Multivariate normality after transformation (usually satisfied) 14 / 24

18 Composite Likelihood (CL) Evaluating the full GP likelihood for 2500 stations is infeasible in a Bayesian model One inversion (or Cholesky decomposition) of the covariance matrix takes seconds CL is an approximation of the full likelihood Stations are broken up into G groups each with n g stations L c (θ y 1,..., y G ) = G L g (θ y g ) (4) g=1 CL Requires O(Gn 3 g) computations as opposed to O(n 3 ). This approximation is applied to the copula as well as each of the GEV parameter residuals. 15 / 24

19 Precipitation Data Global Historical Climatology Network (GHCN), daily total precip data 2500 stations with near complete data from day aggregation window Fall maxima Very large region/dataset for typical Bayesian spatial model Complete Incomplete Knot lon Co Inc 16 / 24

20 Simuion procedure Draw posterior sample Fit the model Compute ent regression coefficient surfaces Unconditional unit Fréchet copula simuion Compute ent parameter means Conditionally simue ent parameter residuals GEV Transformation Latent GEV parameter fields Extreme value simuion! 17 / 24

21 Regression coefficient surfaces - one posterior sample Location Scale Shape Mean seasonal precip lon lon lon Elevation lon lon lon 18 / 24

22 GEV Parameter simuion = conditional sim + mean (covariates + betas) µ(s) = β µ,0 + x T µ(s)β µ (s) + w µ (s) Location Scale Shape lon lon lon 19 / 24

23 Unit Fréchet Copula Simuions lon lon 20 / 24

24 GEV from unit Fréchet If Y is a random variable with a GEV distribution with location µ, scale σ and shape ξ. Then, Z = [1 + ξ(y µ)/σ] 1/ξ is unit Fréchet distributed i.e. Z GEV(1,1,1). If Z is a unit Fréchet random variable. Then, Y = µ + σ(z ξ 1)/ξ is unit GEV distributed with location, scale and shape parameters equal to µ, σ and ξ respectively. 21 / 24

25 Extreme precipitation simuions Precip [mm] sim1 sim2 sim lon Threshold of 1 cm. 22 / 24

26 Discussion and Contributions Discussion: MCMC sampling takes 3+ days! Conditional simuion is expensive Gaussian elliptical copula is an alternative to a max stable process when some conditions hold 23 / 24

27 Discussion and Contributions Discussion: MCMC sampling takes 3+ days! Conditional simuion is expensive Gaussian elliptical copula is an alternative to a max stable process when some conditions hold Conclusions: Bayesian spatial extremes model for return levels Can be used for simuions of extremes on a grid at arbitrary resolution Composite likelihood and spatially varying regression coefficients make it feasible for larger regions 23 / 24

28 Discussion and Contributions Discussion: MCMC sampling takes 3+ days! Conditional simuion is expensive Gaussian elliptical copula is an alternative to a max stable process when some conditions hold Conclusions: Bayesian spatial extremes model for return levels Can be used for simuions of extremes on a grid at arbitrary resolution Composite likelihood and spatially varying regression coefficients make it feasible for larger regions Thanks! 23 / 24

29 References I R A Pielke, M W Downton, and JZB Miller. Flood damage in the United States, : a reanalysis of National Weather Service estimates, F M Ralph, M Dettinger, A White, D Reynolds, D Cayan, T Schneider, R Cifelli, K Redmond, M Anderson, F Gherke, J Jones, K Mahoney, L Johnson, S Gutman, V Chandrasekar, J Lundquist, N Molotch, L Brekke, R Pulwarty, J Horel, L Schick, A Edman, P Mote, J Abatzoglou, R Pierce, and G Wick. A Vision for Future Observations for Western U.S. Extreme Precipitation and Flooding. Journal of Contemporary Water Research & Education, 153(1):16 32, April B Renard. A Bayesian hierarchical approach to regional frequency analysis. Water Resources Research, 47 (W11513):1 21, / 24