Sea-surge and wind speed extremes: optimal estimation strategies for planners and engineers

Abstract

Accurate and precise estimation of return levels is often a key goal of any extreme value analysis. For example, in the UK the British Standards Institution (BSI) incorporate estimates of ‘once-in-50-year wind gust speeds’—or 50-year return levels—into their design codes for new structures; similarly, the Dutch Delta Commission use estimates of the 10,000-year return level for sea-surge to aid the construction of flood defence systems. In this paper, we briefly highlight the shortcomings of standard methods for estimating return levels, including the commonly-adopted block maxima and peaks over thresholds approach, before presenting an estimation framework which we show can substantially increase the precision of return level estimates. Our work allows explicit quantification of seasonal effects, as well as exploiting recent developments in the estimation of the extremal index for handling extremal clustering. From frequentist ideas, we turn to the Bayesian paradigm as a natural approach for building complex hierarchical or spatial models for extremes. Through simulations we show that the return level posterior mean does not have an exceedance probability in line with the intended encounter risk; we also argue that the Bayesian posterior predictive value gives the most satisfactory representation of a return level for use in practice, accounting for uncertainty in parameter estimation and future observations. Thus, where feasible, we propose a Bayesian estimation strategy for optimal return level inference.

Appendices

Appendix 1: extremal index estimators

1.1 Cluster size estimators

• The Runs Estimator: $\hat{\theta }=({\text {mean \,cluster \,size}})^{-1}$, using cluster termination interval $\kappa$ to identify clusters (see Sect. 1.2).

• The blocks estimator: As for the runs estimator, but where blocks of length $\tau$ are considered clusters if there is at least one threshold exceedance within the block.

1.2 Maxima methods

• Gomes’ estimator: Obtain ${(\hat{\mu }_{\theta}}, {\hat{\varsigma }_{\theta}}, {\hat{\xi }_{\theta}})$ for the GEV applied to block maxima $\{M_{\tau }\}$ with block length $\tau$. Find also $(\hat{\mu },\hat{\varsigma },\hat{\xi })$ from block maxima $\{\bar{M}_{\tau }\}$, obtained from an independent series after randomisation of the original series. Then

  $$\begin{aligned} \hat{\theta }\,=\,& {} (\hat{\varsigma }/\hat{\varsigma _{\theta }})^{-1/\tilde{\xi }}, \quad {\text { where}}\\ \tilde{\xi }\,=\,& {} (\hat{\varsigma }-\hat{\varsigma }_{\theta })/(\hat{\mu }-\hat{\mu }_{\theta }) \quad ({\rm Gomes}\,1993). \end{aligned}$$

• Northrop’s estimator:

  $$ \hat{\theta } = -1/\overline{{\text {log }V}}, $$

  with $\overline{\text {log }V} = \sum _{i=1}^{n}{\text {log }}V_{i}/n$, $V_{i}$ being a random sample from a ${\textit{Beta}}(\theta ,1)$ distribution (Northrop 2012).

1.3 Intervals estimators

• Ferro and Segers’ estimator:

  $$ \hat{\theta } = {\text {min}}\left\{ 1, \sum _{i=1}^{J-1}(T_{i}-a)^{2} \big /(J-1)\sum _{i=1}^{J-1}(T_{i}-b)(T_{i}-c)\right\} ,$$

  where $T_{i}=S_{i+1}-S_{i}$, $i=1, \ldots , J-1$ are the times between J threshold exceedances; $a=b=c=0$ if ${\text {max}}(T_{i})\le 2$; otherwise, $a=b=1,c=2$ (Ferro and Segers 2003).

• Süveges’ MLE: Maximum likelihood estimator based on an extension of the work in Ferro and Segers (2003). The likelihood for $U_{i}=T_{i}-1$, $i=1, \ldots , J-1$, is maximised to obtain a closed-form expression for $\hat{\theta }$ (Süveges 2007).

• Süveges’ IWLS: Iterative weighted least squares estimator based on the normalised gaps between clusters (Süveges 2007).

• K -gaps estimator: An extension of Süveges’ MLE, shown to have reduced bias and RMSE (given an optimal choice of tuning parameter K) (Süveges and Davison 2010).

Appendix 2: Bayesian sampling in the hierarchical model

For the hierarchical model outlined in Sect. 3.2, we have

$$\begin{aligned} \eta _{m,s}\,=\,& {} \gamma _{\eta }^{(m)}+\varepsilon _{\eta }^{(s)},\\ \xi _{m,s}\,=\,& {} \gamma _{\xi }^{(m)} + \varepsilon _{\xi }^{(s)} \quad {\text {and}}\\ \alpha _{s}\,=\,& {} \varepsilon _{\alpha }^{(s)} \end{aligned}$$

for the GPD (log) scale and shape, and the logistic dependence parameters (respectively). All random effects for $\eta _{m,s}$ and $\xi _{m,s}$ are assumed to be normally distributed:

$$\begin{aligned} \gamma _{\eta }\sim & {} N(0,\tau ^{-1}_{\eta }) \qquad {\text {and}} \\ \gamma _{\xi }^{(m)}\sim & {} N(0, \tau ^{-1}_{\xi }), \qquad m=1, \ldots , 12, \end{aligned}$$

for seasonal effects, and

$$\begin{aligned} \varepsilon _{\eta }^{(s)}\sim & {} N(a_{\eta },\zeta _{\eta }^{-1}) \qquad {\text {and}} \\ \varepsilon _{\xi }^{(s)}\sim & {} N(a_{\xi }, \zeta ^{-1}_{\xi }), \qquad s=1, \ldots , 12, \end{aligned}$$

for site effects. We choose the mean of the normal distribution of the seasonal effects to be fixed at zero to avoid over-parameterisation and problems of identifiability; however, we could equally have fixed the mean for the distribution of site effects to achieve this. Since the logistic dependence parameter $\alpha$ must lie between 0 and 1, we draw the site effect for $\alpha$ from a uniform distribution, and so

$$ \varepsilon _{\alpha }^{(s)}\sim U(0,1).$$

The final layer of the model is to specify prior distributions for the random effect distribution parameters. Here, we have chosen largely non-informative priors, adopting conjugacy wherever possible to simplify computations. Thus,

$$\begin{aligned}& a_{\eta }\sim N\left( b_{\eta }, c_{\eta }\right) ,\quad a_{\xi } \sim N\left( b_{\xi }, c_{\xi }\right) , \\ &\tau _{\eta }\sim Ga(d_{\eta }, e_{\eta }),\quad \tau _{\xi }\sim Ga(d_{\xi }, e_{\xi }), \qquad {\text {and}}\\ &\zeta _{\eta }\sim Ga(f_{\eta },g_{\eta }),\quad \zeta _{\xi } \sim Ga(f_{\xi }, g_{\xi }), \end{aligned}$$

with a suitable specification of hyper-parameters. The MCMC algorithm employed is Metropolis within Gibbs, i.e. we update each component singly using a Gibbs sampler where the conjugacy allows straightforward sampling from the full conditionals, and a Metropolis step elsewhere. The full conditionals for the Gibbs sampling are:

$$\begin{aligned} a_{\centerdot }|\ldots\sim & {} N\left( \frac{b_{\centerdot }c_{\centerdot } + \zeta _{\centerdot }\sum \varepsilon _{\centerdot }^{(s)}}{c_{\centerdot }+n_{s} \zeta _{\centerdot }},c_{\centerdot }+n_{s}\zeta _{\centerdot }\right) ,\\ \zeta _{\centerdot }|\ldots\sim & {} Ga\left( f_{\centerdot }+\frac{n_{s}}{2}, g_{\centerdot }+\frac{1}{2}\sum (\varepsilon _{\centerdot }^{(s)} - a_{\centerdot })^{2}\right) \end{aligned}$$

and

$$\tau _{\centerdot }|\ldots \sim Ga \left( d_{ \centerdot }+\frac{n_{m}}{2},e_{\centerdot }+\frac{1}{2}\sum (\gamma _{ \centerdot }^{(m)})^{2} \right) ,$$

where $n_{m} {=} {\text {number\, of\, months}} {=} 12$ and $n_{s}$  =  number of sites  =  12, and here the notation $\zeta _{\centerdot}$, for example, is used generically to denote either $\zeta _{\eta }$ or $\zeta _{\xi }$. The complexity of the likelihood derived from the GPD means that conjugacy is unattainable for the random effect parameters, and a Metropolis step is used to update each of these.

In the absence of expert prior knowledge, prior parameters were chosen to give a highly non-informative specification:

$$ b_{\centerdot }=0, c_{\centerdot }=10^{-6}, d_{\centerdot }=e_{\centerdot }=f_{\centerdot }=g_{\centerdot }=10^{-2}. $$