Modelling covariate effects in extremes of storm severityon the Australian North West Shelf
David Randell, Philip Jonathan, Kevin Ewans, Yanyun [email protected]
Shell Technology Centre Thornton, Chester, UK
OMAE Nantes June 2013
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Outline
1 BackgroundMotivationAustralian North West Shelf
2 Modelling CovariatesModel ComponentsP-SplinesQuantile regression models thresholdPoisson models rate of threshold exceedancesGP models size of threshold exceedancesReturn Values
3 Other Applications and Developments
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Contents
1 BackgroundMotivationAustralian North West Shelf
2 Modelling CovariatesModel ComponentsP-SplinesQuantile regression models thresholdPoisson models rate of threshold exceedancesGP models size of threshold exceedancesReturn Values
3 Other Applications and Developments
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Motivation
Rational design an assessment of marine structures:
Reducing bias and uncertainty in estimation of structural reliability.Improved understanding and communication of risk.Climate change.
Other applied fields for extremes in industry:
Corrosion and fouling.Finance.Network traffic.
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Australian North West Shelf
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Australian North West Shelf
Data consist of hindcast storms during 1970-2007.
Model storm peak significant wave height HS .
Wave climate is dominated by westerly monsoonal swell andtropical cyclones.
Cyclones originate from Eastern Indian Ocean and in the Timor andArafura Sea area is also a region of cyclogensis.
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Cyclone Narelle January 2013
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Cyclone Narelle January 2013
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Storm Peak HS by Direction
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Quantiles of storm peak HS Spatially
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Contents
1 BackgroundMotivationAustralian North West Shelf
2 Modelling CovariatesModel ComponentsP-SplinesQuantile regression models thresholdPoisson models rate of threshold exceedancesGP models size of threshold exceedancesReturn Values
3 Other Applications and Developments
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Model Components
Sample {zi}ni=1 of n storm peak significant wave heights observed atlocations {xi , yi}ni=1 with storm peak directions {θi}ni=1.
Model Components1 Threshold function φ above which observations z are assumed to be
extreme estimated using quantile regression.2 Rate of occurrence of threshold exceedances modelled using Poisson
Process model with rate ρ(M= ρ(θ, x , y))
3 Size of occurrence of threshold exceedance using a generalised Pareto(GP) model with shape and scale parameters ξ and σ.
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Model Components
Rate of occurrence and size of threshold exceedance are functionallyindependent (Chavez-Demoulin and Davison 2005).
Equivalent to non-homogeneous Poisson point process model (Dixonet al. 1998).
Smooth functions of covariates are estimated using P-splines (Eilersand Marx 2010)
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P-Splines
Physical considerations suggest that we should expect the modelparameters φ, ρ, ξ and σ to vary smoothly with respect to covariatesθ, x , y .
n dimensional basis matrix B formulated using Kronecker products ofmarginal basis matrices
B = Bθ ⊗ Bx ⊗ By
Roughness is definedR = β′Pβ
where P is penalty matrix formed by taking differences ofneighbouring β.
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P-Splines
Wrapped bases allows forperiodic covariates such asseasonality or direction.
High dimensional bases caneasily be constructed althoughnumber of parametersproblematic.
Strength of roughness penalty iscontrolled by roughnesscoefficient λ: cross validation isused to choose λ optimally.
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Quantile regression models threshold
Estimate smooth quantile φ(θi , xi , yi ; τ) for non-exceedanceprobability τ of storm peak HS .
Spline basis: ψ(τ, θ) =
p∑k=0
Φθkβτk
Estimated by minimising penalised criterion `∗φ with respect to basisparameters:
`∗φ = {τn∑
ri≥0
|ri |+ (1− τ)n∑
ri<0
|ri |}+ λφRφ
for ri = zi − φ(θi , xi , yi ; τ) for i = 1, 2, ..., n, and roughness Rφcontrolled by roughness coefficient λφ.
Quantile regression with P-splines can be formulated and solved as alinear program (Bollaerts et al. 2006).
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Spatio-Directional 50% Quantile Threshold
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Cross Validation for Penalty
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Poisson models rate of threshold exceedances
Rate of occurrence of threshold exceedances is estimated byminimising the roughness penalised log likelihood
`∗ρ = `ρ + λρRρ
(Negative) penalised Poisson log-likelihood for rate of occurrence ofthreshold excesses:
`ρ = −n∑
i=1
log ρ(θi , xi , yi ) +
∫ρ(θ, x , y)dθdxdy
λρ is estimated using cross validation.
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Spatio-Directional Rate of Threshold Exceedances
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GP models size of threshold exceedances
Generalised Pareto density (and negative conditional log-likelihood)for sizes of threshold excesses:
`ξ,σ =n∑
i=1
log σi +1
ξilog(1 +
ξiσi
(zi − φi ))
Parameters: shape ξ, scale σ.
Threshold φi set prior to estimation.
Smoothness is imposed by minimising the roughness penalisedlog-likelihood.
`∗ξ,σ = `ξ,σ + λξRξ + λσRσ
λξ and λσ are estimated using cross validation. In practice setλξ = κλσ for fixed κ.
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Spatio-Directional Scale of GP Exceedances
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Spatio-Directional Shape of GP Exceedances
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Return Values
The return value zT of storm peak significant wave heightcorresponding to some return period T , expressed in years, can beevaluated in terms of estimates for model parameters φ, ρ, ξ and σ
zT = φ− σ
ξ(1 +
1
ρ(log(1− 1
T))−ξ)
z100 corresponds to the 100–year return value, often denoted byHS100.
Return values incorporating effects such as storm dissipation areestimated from simulation.
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Spatio-Directional 100 Year Return Values
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Contents
1 BackgroundMotivationAustralian North West Shelf
2 Modelling CovariatesModel ComponentsP-SplinesQuantile regression models thresholdPoisson models rate of threshold exceedancesGP models size of threshold exceedancesReturn Values
3 Other Applications and Developments
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Other Applications and Developments
Other applications of spline extremes
Seasonal-directionalSpatio-temporal (climate change)
Incorporation of uncertainty
Block bootstrapping allows quick estimates of parameter uncertainty
Incorporation of spatial dependency
Composite likelihood: model (asymptotically dependent)componentwise–maxima.Censored likelihood: allows extension from block-maxima to thresholdexceedances.
Multivariate Extremes
Conditional model for extremes with covariates; Jonathan et al. [2013].
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References
K. Bollaerts, P. H. C. Eilers, and I. Van Mechelen. Simple and multiple p-splinesregression with shape constraints. British Journal of Mathematical &Statistical Psychology, 59:451–469, 2006.
V. Chavez-Demoulin and A.C. Davison. Generalized additive modelling of sampleextremes. J. Roy. Statist. Soc. Series C: Applied Statistics, 54:207, 2005.
J. M. Dixon, J. A. Tawn, and J. M. Vassie. Spatial modelling of extremesea-levels. Environmetrics, 9:283–301, 1998.
P H C Eilers and B D Marx. Splines, knots and penalties. Wiley InterscienceReviews: Computational Statistics, 2:637–653, 2010.
P. Jonathan, K. Ewans, and D. Randell. Joint modelling of extreme oceanenvironments incorporating covariate effects. Coastal Engineering, 97:22–31,2013.
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Spatio-Directional 100-year Return Value HS100
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