generating gridded fields of extreme precipitation for ... · given daily data, if we select the...

Introduction Background BHM Application Simulation References

Generating gridded fields of extreme precipitationfor large domains with a Bayesian hierarchical

model

Cameron BrackenDepartment of Civil, Environmental and Architectural Engineering

University of Colorado at Boulder

Vannes, FranceMay 19, 2016

1 / 24


Coauthors

I Balaji Rajagopalan - University of Colorado

I Will Kleiber - University of Colorado

I Linyin Cheng - NOAA

I Subhrendu Gangopadhyay - Bureau of Reclamation

2 / 24


This talk

I Motivation - Link to weather generation

I Bayesian hierarchical model (BHM) for spatial extremes

I Application to the Western US

I Simulation procedure using the BHM

3 / 24


Motivation - Extremes in the western US?

I From 1983-2000 the WesternStates (WA, OR, CA, ID, NV,UT, AZ, MT, WY, CO, NM,ND, SD, NE, KS, OK, and TX)experienced $24.7 billion inflood damages, $1.5 billionannually.

I California, Washington, andOregon alone accounted for$10.6 billion (46 percent)[Ralph et al., 2014, Pielkeet al., 2002]

Boulder Flood, 2013

Research into hydroclimate extremes can provide moreaccurate and localized estimates of flood risk.

4 / 24


More Motivation

I How are simulated fields of extremes linked to weathergeneration?

I May not need a full weather generator – extremes may be theonly quantity of interest

I Preserve spatial dependence of simulated extremes

I Downscaling

5 / 24


What are BHM Spatial Extremes models typically used for?

I Return Levels

35

40

45

−120 −115 −110 −105lon

lat

1 2 5 13 30 69

100 year returnlevel [cm]

35

40

45

−120 −115 −110 −105lon

lat

1 2 5 13 30 69

100 year returnlevel 90% CI [cm]

6 / 24


Background on Bayesian Statistics

From Bayes’ rule:

p(θ|y, x)︸︷︷︸Posterior

∝ p(y|θ, x)︸︷︷︸Likelihood

p(θ, x)︸︷︷︸Prior

θ Parameters

y Dependent data (response)

x Independent data (covariates/predictors/constants)

Posterior: The answer, probability distributions of parameters

Likelihood: A computable function of the parameters, modelspecific

Prior: Probability distribution, incorporates existingknowledge of the system, model specific

7 / 24


Bayesian Hierarchical Model

In a hierarchical Bayesian model, expand terms using conditionaldistributions where θ = (θ1, θ2):

p(θ |y)︸︷︷︸Posterior

∝ p(y|θ1)︸︷︷︸Data Likelihood

p(θ1|θ2)︸︷︷︸Process Liklihood

p(θ2)︸︷︷︸Prior

Data Likelihood Relates observed data to distributionparameters

Process Likelihood Relates distribution parameters of the toeach other

8 / 24


Statistics of Extremes

Given daily data, if we select the maximum value in each year,those data follow a generalized extreme value (GEV) distribution:

GEV(y; µ, σ, ξ) =1σ

b(−1/ξ)−1 exp{−b−1/ξ

}b = 1 + ξ

(x−µ

σ

), µ: Location, σ: Scale, ξ: Shape.

Return Level (quantile function):

zr = µ+σ

ξ[(− log(1− 1/r))−ξ− 1]

Where r is the return period inyears (100 years for example).

0.0

0.1

0.2

0.3

0.4

-5.0 -2.5 0.0 2.5 5.0y

f(y)

GEV(0,1,-0.5) GEV(0,1,0) GEV(0,1,0.5)

9 / 24


Hierarchical spatial model

(Y(s1, t), . . . , Y(sn, t)) ∼ Cg[Σ; {µ(s), σ(s), ξ(s)}]Y(s, t) ∼ GEV[µ(s), σ(s), ξ(s)]

µ(s) = βµ,0 + xTµ(s)βµ(s) + wµ(s)

σ(s) = βσ,0 + xTσ (s)βσ(s) + wσ(s)

ξ(s) = βξ,0 + xTξ (s)βξ(s) + wξ(s)

wµ(s) ∼ GP(0, C(θµ))

wσ(s) ∼ GP(0, C(θσ))

wξ(s) ∼ GP(0, C(θξ))

Covariates: xT(s) = (Elevation, Mean Seasonal Precip)T

C: Stationary, isotropic, exponential covariance model

10 / 24


Spatially varying regression coefficients

βµ(s) =k

∑i=1

cµi η

µi (s)

βσ(s) =k

∑i=1

cσi ησ

i (s)

βξ(s) =k

∑i=1

cξi η

ξi (s)

ηi(s) = exp(di/φi)

11 / 24


Elliptical copula for data dependence

The Gaussian elliptical copula constructs the joint cdf of of arandom vector (V1, . . . , Vn) as

FC(v1, . . . , vn) = ΦΣ(u1, . . . , un) (1)

where ΦΣ(·) is the joint cdf of an n-dimensional multivariatenormal distribution with covariance matrix Σ, ui = φ−1(Fi[vi]),φ−1 is the inverse cdf (quantile function) of the standard normaldistribution and Fi(·) is the marginal GEV cdf of variable i.

–4 –2 0 2 4–4

–2

0

2

4

Normalized X1

Nor

mal

ized

X2

–4 –2 0 2 4–4

–2

0

2

4

Normalized X1

Nor

mal

ized

X2

0 0.5 10

0.5

1

F1(V1)

F2(V 2

)

–2 0 2 4 6 8

0

500

1000

V1

V 2

Gaussiantransformation

Cumulatedprobability

MultivariateGaussian model

12 / 24


Elliptical copula for data dependenceThe corresponding joint pdf is

fC(y1, . . . , ym) =

m

∏i=1

fi[yi]

m

∏i=1

ψ[ui]

ΨΣ(u1, ....um) (2)

where fi is the marginal GEV pdf at site i, ψ is the standardnormal pdf and ΨΣ is the joint pdf of an m-dimensionalmultivariate normal distribution.

Exponential dependence matrix for spatial data:

Σ(i, j) = exp(−|| si− sj ||/a0) (3)

a0 is the copula range parameter. Termed the dependogram sincevalues are not covariances [Renard, 2011].

13 / 24


Pros and Cons of Gaussian elliptical copulas

Benefits:

I Easy to implement

I Few parameters

I Flexible (can use any marginal distribution)

Requires checking two conditions in application:

I Asymptotic independence (dependence is hard to find inpractice)

I Multivariate normality after transformation (usually satisfied)

14 / 24


Composite Likelihood (CL)

I Evaluating the full GP likelihood for 2500 stations is infeasiblein a Bayesian model

I One inversion (or Cholesky decomposition) of the covariancematrix takes seconds

I CL is an approximation of the full likelihood

I Stations are broken up into G groups each with ng stations

Lc(θ|y1, . . . , yG) =G

∏g=1

Lg(θ|yg) (4)

I CL Requires O(Gn3g) computations as opposed to O(n3).

I This approximation is applied to the copula as well as each ofthe GEV parameter residuals.

15 / 24


Precipitation DataGlobal Historical ClimatologyNetwork (GHCN), daily totalprecip data

I ∼2500 stations with nearcomplete data from1948-2013

I 3 day aggregationwindow

I Fall maxima

Very large region/datasetfor typical Bayesian spatialmodel

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Simulation procedure

Fit the modelDraw posterior sample

Compute latent regression

coefficient surfaces

Compute latent parameter means

Conditionally simulate latent parameter residuals

Unconditional unit Fréchet copula simulation

Extreme value simulation!

Latent GEV parameter fields

GEV Transformation

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Regression coefficient surfaces - one posterior sampleLocation Scale Shape

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Mea

n se

ason

al p

reci

pEl

evat

ion

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GEV Parameter simulation = conditional sim + mean(covariates + betas)

µ(s) = βµ,0 + xTµ(s)βµ(s) + wµ(s)

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0 10 20Location

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0 1 2 3 4 5Scale

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Unit Frechet Copula Simulations

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GEV from unit Frechet

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 Frechet distributed i.e. Z ∼ GEV(1,1,1).

If Z is a unit Frechet random variable. Then,

Y = µ + σ(Zξ − 1)/ξ

is unit GEV distributed with location, scale and shape parametersequal to µ, σ and ξ respectively.

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Extreme precipitation simulations

sim1 sim2 sim3

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2 5 11 26Precip [mm]

Threshold of 1 cm.

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Discussion and Contributions

Discussion:

I MCMC sampling takes 3+ days!

I Conditional simulation is expensive

I Gaussian elliptical copula is an alternative to a max stableprocess when some conditions hold

Conclusions:

I Bayesian spatial extremes model for return levels

I Can be used for simulations of extremes on a grid at arbitraryresolution

I Composite likelihood and spatially varying regressioncoefficients make it feasible for larger regions

Thanks!

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References I

R A Pielke, M W Downton, and JZB Miller. Flood damage in the United States, 1926-2000: a reanalysis ofNational Weather Service estimates, 2002.

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 FutureObservations for Western U.S. Extreme Precipitation and Flooding. Journal of Contemporary Water Research& Education, 153(1):16–32, April 2014.

B Renard. A Bayesian hierarchical approach to regional frequency analysis. Water Resources Research, 47(W11513):1–21, 2011.

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generating gridded fields of extreme precipitation for ... · given daily data, if we select the...

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