Efficient estimation of return value distributions from non...

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Efficient estimation of return value distributions from non-stationarymarginal extreme value models using Bayesian inference

David Randell, Emma Ross, Philip JonathanDavid Randell, EVA 2017

Copyright of Shell David Randell, EVA 2017 June 2017 1 / 28

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South China Sea Storms


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Motivation

Rational and consistent design and assessment of marine structuresReduce bias and uncertainty in estimation of structural integrityQuantify uncertainty as well as possibleNon-stationary marginal, conditional and spatial extremesImproved understanding and communication of riskIncorporation within established engineering design practicesKnock-on effects of improved inference


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South China Sea


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Motivation: storm modelHS ≈ 4× standard deviation of ocean surface time-series at specific location corresponding to a specified period (typically three hours)


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Storm peak data


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Storm peak data by bin

0 45 90 135 180 225 270 315 360

Direction, 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

J

F

M

A

M

J

J

A

S

O

N

D

Sea

son,

(a) Annual rate

0 45 90 135 180 225 270 315 360

0

0.5

1

1.5

2

2.5

3

J

F

M

A

M

J

J

A

S

O

N

D

(b) Maximum


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Storm trajectories

0

30

60

90

120

150

180

210

240

270

300

330

00.511.5


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Marginal: gamma-GP model

Sample of peaks over threshold y, with covariates θθ is 1D in motivating example : directionalθ is nD later : e.g. 4D spatio-directional-seasonal

Below threshold ψy follows truncated gamma with shape α, scale 1/β

Hessian for gamma better behaved than Weibull

Above ψy follows generalised Pareto with shape ξ, scale σ

ξ, σ, α, β, ψ all functions of θψ for pre-specified threshold probability τ

Frigessi et al. [2002], Behrens et al. [2004], MacDonald et al. [2011]


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Gamma-generalised Pareto model for extremes

Density is f(y|ξ, σ, α, β, ψ, τ)

=

τ × fTG(y|α, β, ψ) for y ≤ ψ

(1− τ)× fGP(y|ξ, σ, ψ) for y > ψ

Likelihood is L(ξ, σ, α, β, ψ, τ |yini=1)

=∏

i:yi≤ψ

fTG(yi|α, β, ψ)∏

i:yi>ψ

fGP(yi|ξ, σ, ψ)

× τ nB(1− τ)(1−nB) where nB =∑

i:yi≤ψ

1.

Estimate all parameters as functions of θ


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Marginal: count rate c

Whole-sample rate of occurrence ρ modelled as Poisson process given counts c of numbers ofoccurrences per covariate bin

Chavez-Demoulin and Davison [2005]


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Threshold effect

0 0.2 0.4 0.6 0.8 1

Non-exceedence probability,

-10250

-10000

-9750

-9500

Neg

ativ

e lo

g lik

elih

ood

(a)

0 0.2 0.4 0.6 0.8 1

Non-exceedence probability,

2

3

4

5

6

7

8

1000

0-ye

ar r

etur

n va

lue

(b)


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Threshold effect

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05(a)

2 2.2 2.4 2.6 2.8 3

2

4

6

8

10-4 (b)

0 0.5 1 1.5 2 2.5 3

Storm peak HS

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Den

sity

(c)

2 2.2 2.4 2.6 2.8 3

0.5

1

1.5

2

10-3 (d)


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Marginal: priors and conditional structure

Priors

density of βηκ ∝ exp(−1

2ληκβ

′ηκPηκβηκ

)ληκ ∼ gamma

Conditional structure

f(βη|y,Ω \ βη) ∝ f(y|βη,Ω \ βη)× f(βη|δη,λη)f(λη|y,Ω \ λη) ∝ f(βη|δη,λη)× f(λη)

Ω = α, β, ρ, ξ, σ, ψ, τ

τ is not estimated


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Inference

Elements of βη highly interdependent, correlated proposals essential for good mixing“Stochastic analogues” of IRLS and back-fitting algorithms for maximum likelihood optimisation usedpreviouslyEstimation of different penalty coefficients for each covariate dimension

Gibbs sampling when full conditionals available

Otherwise Metropolis-Hastings (MH) within Gibbs, using suitable proposal mechanismsmMALA where possible

Roberts and Stramer [2002], Girolami and Calderhead [2011], Xifara et al. [2014]


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Posterior parameter estimates

(a)

0 90 180 270 360

JFMAMJJASOND

0.4

0.6

0.8

1

(b)

0 90 180 270 360

JFMAMJJASOND

35

40

45

50

55

(c)

0 90 180 270 360

JFMAMJJASOND

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(d)

0 90 180 270 360

Direction,

JFMAMJJASOND

Sea

son,

0.2

0.4

0.6

0.8

1

1.2

(e)

0 90 180 270 360

JFMAMJJASOND

48

50

52

54

56

58

(f)

0 90 180 270 360

JFMAMJJASOND

-0.06

-0.055

-0.05

-0.045

-0.04


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ValidationCompare sample with simulated values on partitioned covariate domain

NNE

E

SES

SW

W

NW

0 2 4

-2.5

-2

-1.5

-1

-0.5

0A:969, M:1086

0 2 4

-2

-1.5

-1

-0.5

0A:278, M:280

0 2 4-2.5

-2

-1.5

-1

-0.5

0A:360, M:371

0 2 4

-2.5

-2

-1.5

-1

-0.5

0A:457, M:474

0 2 4

-2.5

-2

-1.5

-1

-0.5

0A:840, M:833

0 2 4

-2

-1.5

-1

-0.5

0A:214, M:225

0 1 2

-2

-1.5

-1

-0.5

0A:145, M:136

0 2 4

-3

-2

-1

0A:1368, M:1277

0 1 2 3 4

Storm peak HS

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

log

10(1

-F)

A:4631, M:4682


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Return values

To get directional return values we can do 2 main approachesMonte Carlo simulation: easy to understand and simple to implement but slow. 10000 year eventscan take over a day to compute for the complex models we fit.Numerical integration: much faster, 100 fold improvement in return value calculation time.


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Return values from Monte Carlo simulation

Consider directional-seasonal bin Sj (j = 1, 2, ...,m) centred on location Ij. Sj is sufficiently small thatall model parameters ρ are assumed constant within it.

For each realisation i, for each covariate bin j, with ωj = αj, ζj, ξj, νj, ψj1. Sample the number of storms

nij ∼ Poisson(ρj)

where ρj is the annual rate of occurrence.2. Sample nij ∗ T values from

Yij ∼ GammaGP(ωj)

where T is the return period.

T-year return values in Sj are then found by taking maximum over in each realisation and thenfinding the empirical cdfBins can be combined by taking maximum over bins.


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Numerical integration of return values storm peaks

We define F(y|ωj) to be the cumulative distribution function of any storm peak event given ωj.We estimate the cumulative distribution function FMT

(y|ωj)

FMT(y|ωj) = P (MT < y)

=∞∑k=0

P(k events in Sj in T years

)× Pk (size of an event in Sj < y

)=

∞∑k=0

(Tρj)k

k!exp(−Tρj)× Fk(y|ωj)

= exp(−Tρj

(1− F(y|ωj)

)).


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Posterior predictive return values across bins

Since storm peak events are independent given covariates, we combine by taking the product

FMT(y|ω) =

m∏j=1

FMT(y|ωj)

The final estimate for FMT(y), unconditional on ω, is estimated by marginalising over ω

FMT(y) =

∫ω

FMT(y|ω)f(ω)dω

where f(ω) is the estimated posterior density for ω.


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Empirical dissipation shapes

-90 -45 0 45 900

0.5

1

-90 -45 0 45 900

0.5

1

-90 -45 0 45 900

0.5

1

-90 -45 0 45 900

0.5

1

-90 -45 0 45 900

0.5

1

-90 -45 0 45 90

Angular difference,

0

0.5

1

Pro

babi

lity

-90 -45 0 45 900

0.5

1

-90 -45 0 45 900

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dis

sipa

tion

NNE

E

SES

SW

W

NW


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Numerical integration of return values with dissipation

For applications, it is also necessary to estimate the distribution of return value MTS(y) for maximumof sea state.We empirically estimate the storm dissipation function δ(S; j, y) for sea state HS in directional sector Sestimated from the sample of storm trajectories.Next we estimate the cumulative distribution function FDS(d|ωj) of DS, the dissipated sea state HS insector S from a random storm dissipating from directional-seasonal bin Sj

FDS(d|ωj) = P(DS ≤ d|ωj) =

∫yP(δ(S; j,Y) ≤ d|Y = y)f(y|ωj)dy

where f(y|ωj) is the marginal directional density of storm peak HS in directional-seasonal bin Sj

corresponding to cumulative distribution function F(y|ωj).


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10000 years return values by direction

NNE

E

SES

SW

W

NW

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J FMAMJ J ASOND0

2

4

6

J F M A M J J A S O N D

Season, ?

0

1

2

3

4

5

6

HS


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10000 year return values by season

0 90 180 270 3600

2

4

6Jan

0 90 180 270 3600

2

4

6Feb

0 90 180 270 3600

2

4

6Mar

0 90 180 270 3600

2

4

6Apr

0 90 180 270 3600

2

4

6May

0 90 180 270 3600

2

4

6Jun

0 90 180 270 3600

2

4

6Jul

0 90 180 270 3600

2

4

6Aug

0 90 180 270 3600

2

4

6Sep

0 90 180 270 3600

2

4

6Oct

0 90 180 270 3600

2

4

6Nov

0 90 180 270 3600

2

4

6Dec

0 90 180 270 360

Direction,

0

1

2

3

4

5

6

HS


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Summary

Evidence for covariate effects in marginal extremes of ocean stormsModelling non-stationarity essential for understanding extreme ocean storms, and estimating marine riskwellNon-parametric P-spline flexible basis for covariate descriptionEssential that non-stationary models are used for marginal, conditional and spatial extremes inference ofocean environmentCradle-to-grave uncertainty quantification

Numerical integration of return value provides a much faster way to estimate return values withoutthe need to resort to Monte Carlo simulation.Looking at way of modelling dissipation to avoid the empirical resamplingPaper accepted Ocean Engineering on Monday!http://www.lancs.ac.uk/ jonathan/RssEAMrgBys17.pdf


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References

C N Behrens, H F Lopes, and D Gamerman. Bayesian analysis of extreme events with threshold estimation. StatisticalModelling, 4:227–244, 2004.

V. Chavez-Demoulin and A.C. Davison. Generalized additive modelling of sample extremes. J. Roy. Statist. Soc. SeriesC: Applied Statistics, 54:207–222, 2005.

Arnoldo Frigessi, Ola Haug, and Hå vard Rue. A Dynamic Mixture Model for Unsupervised Tail Estimation withoutThreshold Selection. Extremes, 5:219–235, 2002.

M. Girolami and B. Calderhead. Riemann manifold Langevin and Hamiltonian Monte Carlo methods.J. Roy. Statist. Soc. B, 73:123–214, 2011.

A. MacDonald, C. J. Scarrott, D. Lee, B. Darlow, M. Reale, and G. Russell. A flexible extreme value mixture model.Comput. Statist. Data Anal., 55:2137–2157, 2011.

Gareth O Roberts and Osnat Stramer. Langevin diffusions and metropolis-hastings algorithms. Methodology andcomputing in applied probability, 4(4):337–357, 2002.

T. Xifara, C. Sherlock, S. Livingstone, S. Byrne, and Mark Girolami. Langevin diffusions and the Metropolis-adjustedLangevin algorithm. Stat. Probabil. Lett., 91(2002):14–19, 2014.


Efficient estimation of return value distributions from non...

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