Distributions of return values for ocean wave ...jonathan/SeminarEdinburgh_15.pdfFigure: Directional-seasonal plot for median threshold ensemble rate ~ˆ( 1000) of threshold exceedance

Distributions of return values for oceanwave characteristics usingdirectional-seasonal extreme value analysis

Philip Jonathan, David Randell, Kevin Ewans, Graham FeldShell Stats & Chemometrics

Copyright of Shell Shell Stats & Chemometrics Stats Seminar, Edinburgh January 2015 1 / 47

Acknowledgement

Shell colleagues (statistics, metocean)

Colleagues in academia, especially at Lancaster University

Shell interns and summer students


Motivation: extremes in met-ocean

Rational and consistent design an assessment of marinestructures:

Reduce bias and uncertainty in estimation of return values

Non-stationary marginal and conditional extremes:

Multiple locations, multiple variables, time-seriesMultidimensional covariates

Improved understanding and communication of risk:

Incorporation within well-established engineering designpractices“Knock-on” effects of “improved” inferenceNew and existing structures

Other current applications in Shell:

Geophysics: seismic hazard assessmentAsset integrity: corrosion & fouling


Extremes in met-ocean: univariate challenges

Covariates and non-stationarity:

Location, direction, season, time, water depth, ...Multiple / multidimensional covariates in practice

Cluster dependence:

Same events observed at many locations (pooling)Dependence in time (Chavez-Demoulin and Davison 2012)

Scale effects:

Modelling X or f (X )? (Harris 2004)

Threshold estimation:

Scarrott and MacDonald 2012

Parameter estimation

Measurement issues:

Field measurement uncertainty greatest for extreme valuesHindcast data are simulations based on pragmatic physics,calibrated to historical observation


Extremes in met-ocean: multivariate challenges

Componentwise maxima:⇔ max-stability ⇔ multivariate regular variationAssumes all components extreme⇒ Perfect independence or asymptotic dependence onlyComposite likelihood for spatial extremes (Davison et al. 2012)Point process / multivariate GP process

Extremal dependence: (Ledford and Tawn 1997)Assumes regular variation of joint survivor functionYields more general forms of extremal dependence⇒ Asymptotic dependence, asymptotic independence (with+ve, -ve association), “hidden regular variation”“Ray” extensionsHybrid spatial dependence model (Wadsworth and Tawn 2012)

Conditional extremes: (Heffernan and Tawn 2004)Assumes, given one variable being extreme, convergence ofdistribution of remaining variablesAllows some variables not to be extremeExtensions


Marginal directional-seasonal extremes

googlemaps



Marginal model: single location

Response: storm peak significantwave height, Hsp

S

Wave climate: monsoonal

Southwest monsoon (∼ August, tonorthwest for us)

Northeast monsoon (∼ January, toeast for us)

Long fetches to Makassar Strait,Java Sea

Land shadows of Borneo(northwest), Sulawesi (northeast),Java (south)

www.westernpacificweather.comCopyright of Shell Shell Stats & Chemometrics Stats Seminar, Edinburgh January 2015 7 / 47


Within-storm evolution of significant wave height, HS intime given Hsp

S

Distributions for extreme wave height, crest elevation andsurge given HS

Sample of hindcast storms for period of 1956− 2012

Variables: HS , direction (from, clockwise from north), seasonand wave period information

South China Sea platform (Storm Jangmi, HS = 3.6m,H ≈ 6m): Link


Directional and seasonal variability

Figure: Storm peak significant wave height HspS

(black) on direction θ (upper panel) and season φ (lower panel).

Also shown is sea-state significant wave height HS (grey) on direction θ (upper panel) and season φ (lower panel).Southwest monsoon: August from northwest (315). Northeast monsoon: January from east (90).


Storm model

Figure: HS ≈ 4× standard deviation of ocean surface profile at a location corresponding to a specified period(typically three hours)


Quantiles of H spS

Figure: Empirical quantiles of storm peak significant wave height HspS

by direction θ and season φ, for thresholdnon-exceedance probabilities τ as listed. Empty bins are coloured white.


Storm trajectories of significant wave height, HS

1 2

3

30

210

60

240

90270

120

300

150

330

180

0

Figure: Storm trajectories of significant wave height HS on wave direction θ for 30 randomly-chosen storm events(in different colours). A circle marks the start of each intra-storm trajectory.


Model components

Sample {zi}ni=1 of n storm peak significant wave heightsobserved with storm peak directions {θi}ni=1 and storm peakseasons {φi}ni=1

Model components (all non-stationary w.r.t θ, φ):

1. Threshold function ψu above which observations z areassumed to be extreme estimated using quantile regression

2. Rate of occurrence of threshold exceedances modelled usingPoisson model with rate ρu

3. Size of occurrence of threshold exceedance using generalisedPareto (GP) model with shape and scale parameters ξu and σu

Model estimated for multiple thresholds with non-exceedanceprobabilities τu, u = 1, 2, 3, ...

(Drop sp superscripts and u subscripts where convenient)


Model components

Rate of occurrence and size of threshold exceedancefunctionally independent: (Chavez-Demoulin and Davison2005)

Equivalent to non-homogeneous Poisson point process model

Smooth functions of covariates estimated using penalisedB-splines (Eilers and Marx 2010)

Large number of parameters to estimate:

Slick linear algebra (c.f. generalised linear array models, Currieet al. 2006)Efficient optimisation


Penalised B-splines

Physical considerations suggest model parametersψ, ρ, ξ and σ vary smoothly with covariates θ, φ

Values of (η =)ψ, ρ, ξ and σ all take the form:

η = Bβη

for B-spline basis matrix B (defined on index set of covariatevalues) and some βη to be estimated

Multidimensional basis matrix B formulated using Kroneckerproducts of marginal basis matrices:

B = Bθ ⊗ Bφ

(exact operations calculated without explicit evaluation)

Roughness Rη defined as:

Rη = β′ηPβη

where effect of P is to difference neighbouring values of βη


Penalised B-splines

Wrapped bases for periodiccovariates (seasonal,direction)

Multidimensional baseseasily constructed. Problemsize sometimes prohibitive

Parameter smoothnesscontrolled by roughnesscoefficient λ: crossvalidation or similarchooses λ optimally

Alternatives: random fields,Gaussian processes, ...


Quantile regression for extreme value threshold

Estimate smooth quantile ψ(θ, φ; τ) for non-exceedanceprobability τ of z (storm peak HS) using quantile regressionby minimising penalised criterion `∗ψ with respect to basisparameters:

`∗ψ = `ψ + λψRψ

`ψ = {τn∑

ri≥0

|ri |+ (1− τ)n∑

ri<0

|ri |}

for ri = zi − ψ(θi , φi ; τ) for i = 1, 2, ..., n, and roughness Rψcontrolled by roughness coefficient λψ

(Non-crossing) quantile regression formulated as linearprogramme (Bollaerts et al. 2006)

λψ estimated using cross validation or similar


Directional-seasonal extreme value threshold, ψ

τ=0.5

0 90 180 270 360

JFMAMJJASOND

τ=0.55

0 90 180 270 360

JFMAMJJASOND

τ=0.6

0 90 180 270 360

JFMAMJJASOND

τ=0.65

0 90 180 270 360

JFMAMJJASOND

τ=0.7

0 90 180 270 360

JFMAMJJASOND

τ=0.75

0 90 180 270 360

JFMAMJJASOND

Direction

Sea

son

τ=0.8

0 90 180 270 360

JFMAMJJASOND

τ=0.85

0 90 180 270 360

JFMAMJJASOND

τ=0.9

0 90 180 270 360

JFMAMJJASOND

HSsp

0.5

1

1.5

2

2.5

Figure: Directional-seasonal plots for extreme value thresholds, ψ, corresponding to equally spaced non-exceedanceprobabilities of H

spS

on [0.5, 0.9] (left to right, then top to bottom).


Accommodating multiple thresholds

Median threshold ensemble estimates

ρ = medu

{ρu

τu1− τu

}σ = med

u{σu + ξu(ψu − ψu)}

ξ = medu{ξu}

Parameter estimates can be fairly compared

τu set to 0.5


Poisson model for rate of threshold exceedance

Poisson model for rate of occurrence of threshold exceedanceestimated by minimising roughness penalised log likelihood:

`∗ρ = `ρ + λρRρ

(Negative) penalised Poisson log-likelihood (andapproximation):

`ρ = −n∑

i=1

log ρ(θi , φi ) +

∫ρ(θ, φ)dθdφ

ˆρ = −

m∑j=1

cj log ρ(j∆) + ∆m∑j=1

ρ(j∆)

{cj}mj=1 counts of threshold exceedances on index set of m(>> 1) bins partitioning covariate domain into intervals ofvolume ∆

λρ estimated using cross validation or similar


Directional-seasonal exceedance rate, ρ

0 90 180 270 3600

2

4

Jan

0 90 180 270 3600

2

4

Feb

0 90 180 270 3600

2

4

Mar

0 90 180 270 3600

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4

Apr

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4

Sep

0 90 180 270 3600

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4

Nov

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2

4

Dec

Direction

Sea

son

Rate

0 90 180 270 360

J

F

M

A

M

J

J

A

S

O

N

D

1 2 3 4

Figure: Directional-seasonal plot for median threshold ensemble rate ρ(×1000) of threshold exceedance of HspS

.

The left-hand panel shows ρ(×1000) on θsp and φsp . The right hand panel shows 12 monthly directionalestimates with 95% BCa bootstrap confidence intervals (dashed).


GP model for size of threshold exceedance

Generalise Pareto model for size of threshold exceedanceestimated by minimising roughness penalised log-likelihood:

`∗ξ,σ = `ξ,σ + λξRξ + λσRσ

(Negative) conditional generalised Pareto log-likelihood:

`ξ,σ =n∑

i=1

log σi +1

ξilog(1 +

ξiσi

(zi − ψi ))

Parameters: shape ξ, scale σ

Threshold ψ set prior to estimation

λξ and λσ estimated using cross validation or similar. Inpractice set λξ = κλσ for fixed κ


Directional-seasonal parameter plot for GP scale, σ

0 90 180 270 3600

0.2

0.4

0.6

0.8Jan

0 90 180 270 3600

0.2

0.4

0.6

0.8Feb

0 90 180 270 3600

0.2

0.4

0.6

0.8Mar

0 90 180 270 3600

0.2

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0.2

0.4

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0.8Dec

Direction

Sea

son

Scale

0 90 180 270 360

J

F

M

A

M

J

J

A

S

O

N

D

0.1 0.2 0.3 0.4 0.5

Figure: Directional-seasonal plot for median threshold ensemble generalised Pareto scale, σ. The left-hand panelshows σ on θ and φ. The right hand panel shows 12 monthly directional estimates with 95% BCa bootstrapconfidence intervals (dashed).


Directional-seasonal parameter plot for GP shape, ξ

0 90 180 270 360−0.2

−0.1

0

0.1

Jan

0 90 180 270 360−0.2

−0.1

0

0.1

Feb

0 90 180 270 360−0.2

−0.1

0

0.1

Mar

0 90 180 270 360−0.2

−0.1

0

0.1

Apr

0 90 180 270 360−0.2

−0.1

0

0.1

May

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−0.1

0

0.1

Jun

0 90 180 270 360−0.2

−0.1

0

0.1

Jul

0 90 180 270 360−0.2

−0.1

0

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Aug

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−0.1

0

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Oct

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−0.1

0

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Sep

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−0.1

0

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Nov

0 90 180 270 360−0.2

−0.1

0

0.1

Dec

Direction

Sea

son

Shape

0 90 180 270 360

J

F

M

A

M

J

J

A

S

O

N

D

−0.1 −0.05 0 0.05

Figure: Directional-seasonal plot for median over threshold generalised Pareto shape, ξ. The left-hand panel showsξ on θ and φ. The right hand panel shows 12 monthly directional estimates with 95% BCa bootstrap confidenceintervals (dashed).


Return values

Estimation of return values by simulation under the modelThreshold level selected at randomNumber of events in periodDirections and seasons of each eventSize (or magnitude) of each eventHS100 is the maximum value of Hsp

S in a simulation period of100–years

Alternative: closed form function of parametersReturn value zT of storm peak significant wave heightcorresponding to return period T (years) evaluated fromestimates for ψ, ρ, ξ and σ:

zT = ψ − σ

ξ(1 +

1

ρ(log(1− 1

T))−ξ)

Implementation and interpretation problematic


Accommodating multiple thresholds

Threshold ensemble estimates of return value distributions

Pr(Q ≤ x) =

∫τ∈Jτ

Pr(Q ≤ x |τ)dF (τ)

≈ 1

nτ

nτ∑u=1

Pr(Q ≤ x |τu)

The quantiles q(p) are solutions to Pr(Q ≤ x) = p.

Incorporates threshold variability in return value estimate


Return value plot for HS100, q(0.5)

0 90 180 270 3600

2

4

Jan

0 90 180 270 3600

2

4

Feb

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2

4

Mar

0 90 180 270 3600

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4

Apr

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Jul

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Aug

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4

Oct

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2

4

Sep

0 90 180 270 3600

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4

Nov

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2

4

Dec

Direction

Sea

son

HS

0 45 90 135 180 225 270 315

J

F

M

A

M

J

J

A

S

O

N

D

0 1 2 3

Figure: Directional-seasonal return value plot for 100-year significant wave height (in metres). The left-hand panelshows directional and seasonal variability of the median threshold ensemble estimate q(0.5) for HS . The right handpanel shows 12 monthly directional octant return values (in black) in terms of BCa 95% confidence limits forq(0.5) (solid), q(0.025) (dashed) and q(0.975) (dashed). Also shown are the corresponding omni-directionalestimates (in red).


Within-storm variability

Figure: Wave has removed boat landing gear


Within-storm variability

Critical environmental variables:

Storm peak significant wave height:

(Sea state) significant wave heightMaximum wave heightMaximum crest elevation, CPeak total water level (≈ crest + surge + tide)

“Associated” values of wind speed and directioncorresponding to peak significant wave height:

Maximum conditional structural loads and responsesConditional extremes


Estimating within-storm variability

Extreme value model allows simulation of HspS , θsp and φsp

Matching procedure used to estimate storm evolution(HS(t), θ(t), φ(t))|(Hsp

S , θsp, φsp) for sea state t

Essential in estimating return values for covariate bins otherthan that containing the storm peakOpportunity for empirical modelling

Empirical (physics-motivated) literature models for C |HS(t)

The cumulative distribution function for the maximum crest elevation C in a sea-state parameterised by S of nSwaves with significant wave height HS = hS is taken (see, e.g. Forristall 1978, 2000) to be given by:

Pr(C ≤ η|S) = (1− exp(−η

αShS)βS )nS

where all of αS , βS and nS are functions of the sea-state parameters S estimated from observation.


Directional-seasonal return value plot for C100

0 90 180 270 3600

2

4

6Jan

0 90 180 270 3600

2

4

6Feb

0 90 180 270 3600

2

4

6Mar

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6Oct

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6Sep

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4

6Nov

0 90 180 270 3600

2

4

6Dec

Direction

Sea

son

C

0 45 90 135 180 225 270 315

J

F

M

A

M

J

J

A

S

O

N

D

0 1 2 3 4

Figure: Directional-seasonal return value plot for 100-year crest elevation (in metres). The left-hand panel showsdirectional and seasonal variability of the median quantile over threshold q(0.5) for C . The right hand panel shows12 monthly directional octant return values (in black) in terms of BCa 95% confidence limits for q(0.5) (solid),q(0.025) (dashed) and q(0.975) (dashed). Also shown are the corresponding omni-directional estimates (in red).


Validation of model for sea-state HS

0 1 2 3 4−5

−4

−3

−2

−1

0Jan (8528, 8279)

0 1 2 3 4−5

−4

−3

−2

−1

0Feb (8627, 7974)

0 1 2 3 4−5

−4

−3

−2

−1

0Mar (8734, 7948)

0 1 2 3 4−5

−4

−3

−2

−1

0Apr (9175, 8670)

0 1 2 3 4−5

−4

−3

−2

−1

0May (9069, 8650)

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−4

−3

−2

−1

0Jun (8815, 8437)

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−4

−3

−2

−1

0Jul (9137, 8715)

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0Aug (9046, 8560)

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−2

−1

0Oct (9091, 8964)

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−1

0Sep (9274, 8257)

0 1 2 3 4−5

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−1

0Nov (8996, 8003)

0 1 2 3 4−5

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−2

−1

0Dec (8969, 7560)

0 1 2 3 4−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0Omni (107461, 100016)

HS

log(

1−F

(HS))

Figure: Illustration of validation of return value estimation for significant wave height by comparison of cumulativedistribution functions (cdfs) for 1000 bootstrap resamples of the original sample with those from 1000 samplerealisations under the model (incorporating intra-storm evolution of HS ) corresponding to the same time period asthe original sample. The 12 right hand panels show empirical 95% bootstrap uncertainty bands for monthlyomni-directional cdfs for the original sample (red), and BCa 95% confidence intervals for the 2.5%ile and 97.5%ilemedian over threshold estimates q(0.025) and q(0.975) (both dashed). Titles for plots, in brackets following themonth name, are the numbers of actual and simulated events in each month. The left hand panel makes theequivalent omni-directional, omni-seasonal comparison.


Interval of threshold non-exceedance probability

0.3 0.5 0.7 0.9

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3

4

5

Jan

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Dec

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2

2.5

3

3.5

4

4.5

5

Omni

τ

HS

Figure: Estimates for 100-year maximum for HspS

from simulation under models corresponding to 100 bootstrapresamples for each of 15 choices of threshold non-exceedance probability, τ . Median estimates are connected by asolid red line. 2.5% and 97.5% iles are connected by dashed red lines. The left hand panel shows theomni-directional, omni-seasonal estimate. The right hand panels show 12 monthly omni-directional estimates.


Bootstrap threshold ensemble, Q

Bootstrap threshold ensemble return value Q

Estimate return value distribution Q|B, τ by simulation forthreshold non-exceedance probability τ and bootstrapresample B ∈ B

Pr(Q ≤ x) =

∫τ∈Jτ

∫B∈B

Pr(Q ≤ x |B, τ)dF (B)dF (τ)

≈ 1

nτ

1

nB

nτ∑u=1

nB∑b=1

Pr(Q ≤ x |Bb, τu)


Bootstrap threshold ensemble Q and median threshold ensemble Q

1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Directional threshold ensemble with BCa confidence intervals

1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

HS

Cum

ulat

ive

prob

abili

ty

Directional bootstrap threshold ensemble

[−22.5,22.5][22.5,67.5][67.5,112.5][112.5,157.5][157.5,202.5][202.5,247.5][247.5,292.5][292.5,337.5]Omni

1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Seasonal threshold ensemble with BCa confidence intervals

1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Seasonal bootstrap threshold ensemble

JanFebMarAprMayJunJulAugSepOctNovDecOmni

Figure: Empirical cumulative distribution functions (cdfs) for 100-year significant wave height from simulationunder the directional-seasonal model. Left hand and right hand panels show directional and seasonal cdfsrespectively. Upper panels shows median threshold ensemble estimates Q with 95% BCa confidence intervals, andlower panels bootstrap threshold ensemble estimates Q.


Non-stationary extremes: developments

Marginal models:Other covariate representationsExtension to higher-dimensional covariates

Computational efficiency:More sparse and slick matrix manipulations, optimisationParallel implementation

Bayesian formulation

Spatial model:Composite likelihood: model componentwise maximaNon-stationary dependenceCensored likelihood: block maxima → threshold exceedancesHybrid model: mix AD and AI?

Non-stationary conditional extremes:Multidimensional covariatesMultivariate response

Incorporation within structural design framework


References

K Bollaerts, P H C Eilers, and M Aerts. Quantile regression with monotonicity restrictions using P-splines and theL1 norm. Statistical Modelling, 6:189–207, 2006.

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

V. Chavez-Demoulin and A.C. Davison. Modelling time series extremes. REVSTAT - Statistical Journal, 10:109–133, 2012.

I. D. Currie, M. Durban, and P. H. C. Eilers. Generalized linear array models with applications to multidimensionalsmoothing. J. Roy. Statist. Soc. B, 68:259–280, 2006.

A. C. Davison, S. A. Padoan, and M. Ribatet. Statistical modelling of spatial extremes. Statistical Science, 27:161–186, 2012.

P H C Eilers and B D Marx. Splines, knots and penalties. Wiley Interscience Reviews: Computational Statistics, 2:637–653, 2010.

G. Z. Forristall. On the statistical distribution of wave heights in a storm. J. Geophysical Research, 83:2353–2358,1978.

G. Z. Forristall. Wave crest distributions: Observations and second-order theory. Journal of PhysicalOceanography, 30:1931–1943, 2000.

R. I. Harris. Extreme value analysis of epoch maxima-convergence, and choice of asymptote. Journal of WindEngineering and Industrial Aerodynamics, 92:897–918, 2004.

J. E. Heffernan and J. A. Tawn. A conditional approach for multivariate extreme values. J. R. Statist. Soc. B, 66:497–546, 2004.

P. Jonathan, D. Randell, Y. Wu, and K. Ewans. Return level estimation from non-stationary spatial data exhibitingmultidimensional covariate effects. Ocean Eng., 88:520–532, 2014.

A. W. Ledford and J. A. Tawn. Modelling dependence within joint tail regions. J. R. Statist. Soc. B, 59:475–499,1997.

C. Scarrott and A. MacDonald. A review of extreme value threshold estimation and uncertainty quantification.REVSTAT - Statistical Journal, 10:33–60, 2012.

J.L. Wadsworth and J.A. Tawn. Dependence modelling for spatial extremes. Biometrika, 99:253–272, 2012.


Marginal spatio-directional

Figure: Hurricane Katrina


Marginal spatio-directional

Longitude, latitude and direction as covariates

Physics: direction and season correlatedGulf of Mexico (GoM), North West Shelf of Australia (NWS)applications here

Marginal per location

Estimation of spatial smoothness

Sample is spatially dependentVertical adjustment / sandwich estimator(Spatial) block bootstrap


GoM spatio-directional H spS

Figure: ≈ 17000 locations × 32 directional bins for Gulf of Mexico. Plot forquantile (withheld) of 100-year maximum storm peak significant waveheight, Hsp

S


NWS spatio-directional H spS

Figure: North West Shelf of Australia. See Jonathan et al. [2014]


Non-stationary conditional extremes

Figure: Floating LNG tanker (500m long!)



Problem structure:

Bivariate sample {xij}n,2i=1,j=1 of random variables X1, X2

Covariate values {θij}n,2i=1,j=1 associated with each individual

For some choices of variables X , e.g. X1 = HS , X2 = TP ,θi1 , θi2

For other choices, e.g. X1 = HS , X2 =WindSpeed, θi1 6= θi2in general

We will assume θi1 = θi2 = θi

Objective:

Objective: model the joint distribution of extremes of X1 andX2 as a function of θ

(Drop subscripts wherever possible for convenience)Copyright of Shell Shell Stats & Chemometrics Stats Seminar, Edinburgh January 2015 43 / 47


On Gumbel scale, by analogy with Heffernan and Tawn [2004] wepropose the following conditional extremes model:

(Xk |Xj = xj , θ) = αθxj + xβθj (µθ + σθZ ) for xj > φGjτ ′(θ)

where:

φGjτ ′(θ) is a high directional quantile of Xj on Gumbel scale,above which the model fits well

αθ ∈ [0, 1], βθ ∈ (−∞, 1], σθ ∈ [0,∞)

Z is a random variable with unknown distribution G

Z will be assumed to be approximately Normally distributedfor the purposes of parameter estimation

αθ, βθ, µθ and σθ are functions of direction with B-splineparameterisations


North Sea marginal return values for TP (simulation)

Figure: Omni-directional and sector marginal distributions of 100-year TspP


North Sea conditional return values (simulation)

Figure: Omni-directional and sector conditional distributions of storm peak period, TspP

given 100-year HspS

usingextension of model of Heffernan & Tawn incorporating non-stationarity



Documents

Distributions of return values for ocean wave ...jonathan/SeminarEdinburgh_15.pdfFigure: Directional-seasonal plot for median threshold ensemble rate ~ˆ( 1000) of threshold exceedance