International Graduate Summer School on Statistics and Climate Modeling, Boulder, CO, 8/12/08

Case Study II: Extremes

Markov Random Fields and Regional Climate Models

Stephan R. SainGeophysical Statistics Project

Institute for Mathematics Applied to Geosciences

National Center for Atmospheric Research

Boulder, CO

Supported by NSF ATM/DMS. Thanks also to Dan Cooley.

Goals

2

• How to model spatial extremes?

• Can we model GEV parameters spatially? In particular the

shape parameter?

• What kinds of changes are projected for extreme precipita-

tion?

Extreme Value Distributions

3

• Let Z denote a block maxima (e.g. annual). Under certain

conditions, it can be assumed that Z follows a generalized

extreme value distribution given by

G(z) = exp

[−

(1 + ξ

z − µ

σ

)−1/ξ]

,

for values of z such that 1 + ξ(z − µ)/σ > 0.

– µ and σ are location and scale parameters.

– If the shape parameter ξ < 0, then the distribution has a

bounded upper tail. If ξ = 0 (taken as the limit), then

the distribution has an exponentially decreasing tail, and,

if ξ > 0, the distribution’s tail decays as a power function.

Extreme Value Distributions

4

• The block maxima approach is limited by using only one data

point per block; alternative approaches include models for all

points over a threshold.

– Generalized Pareto Distribution (GPD)

– Point processes

• From the GEV or GPD, extreme behavior of the distribution

can be summarized through return levels or, more simply, val-

ues of extreme quantiles implied from the distribution.

A Hierarchical Model for Extremes

5

• For each grid box, let Zr,i,t denote the precipitation amount

recorded for the rth simulation (control or future), the ith

grid cell, and the tth day.

• Assume that all Zr,i,t the exceed the threshold ur,i follow the

point process model with parameters µr,i, σr,i, and ξr,i and are

conditionally independent of other grid boxes.

• Note: the likelihood for the point process model augmented

by the regularization/prior suggested by Martins and Stedinger

(2000).

A Hierarchical Model for Extremes

6

• In the process model, the parameters are modeling via

µr,i ∼ N(X ′iβr,µ + Ur,i,µ,1/τ2

µ)

log(σr,i) ∼ N(X ′iβr,σ + Ur,i,σ,1/τ2

σ )

ξr,i ∼ N(X ′iβr,ξ + Ur,i,ξ,1/τ2

ξ )

– Regression includes elevation, longitude, latitude, ocean,

but spatial patterns assumed to be the same across control

and future (different intercepts).

– U = (U1,µ,U1,σ,U1,ξ,U2, µ,U2,σ,U2,ξ)′ represents a spatial

random effect modeled through an intrinsic autoregressive

process (IAR) with precision matrix Q = T⊗W.

Intrinsic Autogressive Process

7

• The IAR is an improper version of the CAR model.

• T is a 6× 6 covariance matrix.

• W has:

– non-diagonal elements wi,j = −1 if grid boxes i and j are

neighbors and zero otherwise.

– wi,i = −∑

j 6=i wi,j

• All row sums of W are zero, imposing the conditions∑i Ur,i,θ = 0 for r = 1,2 and all θ = µ, σ, ξ.

A Hierarchical Model for Extremes

8

• Conjugate priors for βs (Gaussian) and T (Whishart) with

ranges of variation reflective of exploratory analysis (i.e. MLE).

• Gibbs sampler used to sample from the posterior, with special

care taken for the spatial random effect, U.

A Comparison of ξ

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3035

4045

5055

longitude

latit

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−0.

2−

0.1

0.0

0.1

0.2

0.3

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−0.

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0.0

0.1

0.2

0.3

Winter Model Parameters

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020

4060

80

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longitudela

titud

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1020

3040

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ude

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

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020

4060

80

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1020

3040

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longitudela

titud

e

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Winter Model Parameters

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−4

−2

02

4

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3540

4550

55

longitude

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ude

−6

−4

−2

02

46

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−0.

050.

000.

05

Winter Return Levels

12

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5010

015

020

025

0

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5010

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020

025

0

Winter Return Levels

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−30

−20

−10

010

2030

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0.0

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Winter Spatial Effect

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020

4060

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titud

e

−1.

0−

0.5

0.0

0.5

1.0

1.5

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latit

ude

−0.

06−

0.04

−0.

020.

000.

020.

040.

06

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longitude

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ude

−20

020

4060

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ude

−1.

0−

0.5

0.0

0.5

1.0

1.5

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titud

e

−0.

06−

0.04

−0.

020.

000.

020.

04

Summer Return Levels

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4060

8010

012

014

0

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ude

020

4060

8010

012

014

0

Summer Return Levels

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ude

−50

050

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0.0

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1.0

A Big Issue

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