Joint modelling of extreme ocean environments incorporating covariate effects Philip Jonathan Shell Research Ltd., Chester, CH1 3SH, UK. Kevin Ewans Sarawak Shell Bhd., 50450 Kuala Lumpur, Malaysia. David Randell Shell Research Ltd., Chester, CH1 3SH, UK. Abstract Characterising the joint distribution of extremes of ocean environmental variables such as significant wave height (H S ) and spectral peak period (T P ) is important for understanding extreme ocean environments and in the design and assessment of marine and coastal structures. Many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between vari- ables without justification. Models are also typically restricted to joint regions in which all variables are extreme, but regions where only a subset of variables are extreme can be equally important for design. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these difficulties. Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection, marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter uncertainty with respect to covariate. 1
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Joint modelling of extreme ocean environments incorporating covariate effects
Philip Jonathan
Shell Research Ltd., Chester, CH1 3SH, UK.
Kevin Ewans
Sarawak Shell Bhd., 50450 Kuala Lumpur, Malaysia.
David Randell
Shell Research Ltd., Chester, CH1 3SH, UK.
Abstract
Characterising the joint distribution of extremes of ocean environmental variables such as significant wave
height (HS) and spectral peak period (TP ) is important for understanding extreme ocean environments
and in the design and assessment of marine and coastal structures. Many applications of multivariate
extreme value analysis adopt models that assume a particular form of extremal dependence between vari-
ables without justification. Models are also typically restricted to joint regions in which all variables are
extreme, but regions where only a subset of variables are extreme can be equally important for design.
The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these
difficulties.
Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection,
marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent
extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised
likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with
cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter
uncertainty with respect to covariate.
1
Philip.Jonathan
Typewritten Text
Published. Coastal Engineering 79 (2013) 22–31.
We illustrate the approach in application to joint modelling of storm peak HS and TP at a Northern North
Sea location with storm direction as covariate. We evaluate the impact of incorporating directional effects
on estimates for return values, including those of a structure variable, similar to the structural response of
a floating structure. We believe the approach offers the ocean engineer a straightforward procedure, based
on sound statistics, to incorporate covariate effects in estimation of joint extreme environmental conditions.
1 Figure 1: North Sea location. Directional sectors corresponding to long fetches associatedwith the Atlantic Ocean, Norwegian Sea and North Sea typically yield more severe stormevents. Sectors corresponding to Norway and the United Kingdom are fetch limited. Stormdirection is the direction from which the storm emanates, and is measured clockwise fromNorth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Plots of storm peak HS (in metres, horizontal) versus associated TP (in seconds, vertical) forthe most severe (20% of) storms emanating from 6 directional sectors (ordered, clockwisefrom 20o). The characteristics of dependence between TP and HS varies from sector tosector. For example, for storms emanating from the south (directional sector [140, 200)), TPis highly dependent on HS in contrast to Atlantic storms (from directional sector [230, 280)). 26
3 Polar plot of marginal directional quantile estimates for storm peak HS (in metres, onleft hand side) and TP (seconds) for deciles with probabilities 0.1 to 0.8 together with thesample (grey dots). Transformed directions were used for quantile regression. Directionaldependence is apparent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Estimates for parameters α, β, µ and σ and their uncertainties as functions of θ usingpenalised likelihood with bootstrap resampling. Median bootstrap estimate is given in solidblack, with a 95% bootstrap uncertainty band in dashed black. The estimate obtained usingthe original sample is given in solid grey. Dependence term α is largest for storms emanatingfrom the North Sea sector as expected from inspection of sample (see Figure 2). Directionaleffects are apparent in α and σ in particular. The bootstrap uncertainty in β is particularlylarge. Note that parameter uncertainty is also large in the interval [0,135) (not shown), dueto small sample size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Marginal storm peak TP (seconds) corresponding to non-exceedance probability of 0.99(in 34 years), incorporating covariate effects (solid light grey). Median conditional stormpeak TP given exceedances of storm peak HS (metres) with exceedance probability 0.01,incorporating (solid black) and ignoring (solid dark grey) covariate effects, together with2.5%- and 97.5% percentiles (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Return values of storm peak HS (metres) and associated conditional values of TP (seconds).Inner dot-dashed lines (on common scale): storm peak HS with probability non-exceedanceprobability 0.99 (in 34 years), with (grey) and without (black) directional effects. Outer solidlines (on common scale): median associated TP with (black) and without (grey) directionalcovariate effects; outer dashed lines give corresponding 2.5%- and 97.5% percentile valuesfor associated TP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7 Simple ratio of response R to storm peak HS as a function of storm peak TP (seconds) forany direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Values for residuals Z from conditional extremes model against direction θ. There is noobvious directional dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10 Values for residuals Z from conditional extremes model against conditioning variate HS (onGumbel scale). There is no obvious dependence. . . . . . . . . . . . . . . . . . . . . . . . . 34
23
11 Values for residuals Z from a covariate-free conditional extremes model against direction θ.Residuals appear to be dependent on the value of covariate. . . . . . . . . . . . . . . . . . . 35
24
Figure 1: Figure 1: North Sea location. Directional sectors corresponding to long fetches associated with the Atlantic Ocean,Norwegian Sea and North Sea typically yield more severe storm events. Sectors corresponding to Norway and the UnitedKingdom are fetch limited. Storm direction is the direction from which the storm emanates, and is measured clockwise fromNorth.
25
Figure 2: Plots of storm peak HS (in metres, horizontal) versus associated TP (in seconds, vertical) for the most severe (20%of) storms emanating from 6 directional sectors (ordered, clockwise from 20o). The characteristics of dependence between TP
and HS varies from sector to sector. For example, for storms emanating from the south (directional sector [140, 200)), TP ishighly dependent on HS in contrast to Atlantic storms (from directional sector [230, 280)).
26
Figure 3: Polar plot of marginal directional quantile estimates for storm peak HS (in metres, on left hand side) and TP
(seconds) for deciles with probabilities 0.1 to 0.8 together with the sample (grey dots). Transformed directions were used forquantile regression. Directional dependence is apparent.
27
Figure 4: Estimates for parameters α, β, µ and σ and their uncertainties as functions of θ using penalised likelihood withbootstrap resampling. Median bootstrap estimate is given in solid black, with a 95% bootstrap uncertainty band in dashedblack. The estimate obtained using the original sample is given in solid grey. Dependence term α is largest for stormsemanating from the North Sea sector as expected from inspection of sample (see Figure 2). Directional effects are apparentin α and σ in particular. The bootstrap uncertainty in β is particularly large. Note that parameter uncertainty is also largein the interval [0,135) (not shown), due to small sample size.
28
Figure 5: Marginal storm peak TP (seconds) corresponding to non-exceedance probability of 0.99 (in 34 years), incorporatingcovariate effects (solid light grey). Median conditional storm peak TP given exceedances of storm peak HS (metres) withexceedance probability 0.01, incorporating (solid black) and ignoring (solid dark grey) covariate effects, together with 2.5%-and 97.5% percentiles (dashed).
29
Figure 6: Return values of storm peak HS (metres) and associated conditional values of TP (seconds). Inner dot-dashedlines (on common scale): storm peak HS with probability non-exceedance probability 0.99 (in 34 years), with (grey) andwithout (black) directional effects. Outer solid lines (on common scale): median associated TP with (black) and without(grey) directional covariate effects; outer dashed lines give corresponding 2.5%- and 97.5% percentile values for associated TP .
30
Figure 7: Simple ratio of response R to storm peak HS as a function of storm peak TP (seconds) for any direction.
31
Figure 8: Median conditional structural response (in metres) for storm peak HS (metres) exceeding its directional quantilenon-exceedance probability of 0.99 incorporating (solid black) and ignoring (solid grey) directional covariate effect, withcorresponding 2.5%- and 97.5% percentiles (dashed).
32
Figure 9: Values for residuals Z from conditional extremes model against direction θ. There is no obvious directionaldependence.
33
Figure 10: Values for residuals Z from conditional extremes model against conditioning variate HS (on Gumbel scale). Thereis no obvious dependence.
34
Figure 11: Values for residuals Z from a covariate-free conditional extremes model against direction θ. Residuals appear tobe dependent on the value of covariate.