Suppose that we have an estimate $θ of the complete vector of parameters introduced in Section 4.5, and a measure of its uncertainty, obtained for instance using a block jackknife or bootstrap, and that we wish to check the quality of the model. For each sampled location s1, . . . , sL, we can compare the observations with the theoretical
quantiles of the marginal model. Let uq(sl) denote the qt hempirical quantile of r -
exceedances at location sl, i.e., estimated using only observations satisfying r (Xn)u,
and let nqdenote the number of observations exceeding uq(sl). Following equation
(4.13), we have
Pr X (sl)− uq(sl)x| X (sl)uq(sl)
≈ H$ξ,$σ(sl)(x), x0,
withσ(s$ l)= 2a(n) $A(sl)+ $ξ{uq(sl)− $B (sl)− 2b(n)}. Then we use quantile-quantile plots
to check the quality of the marginal fit. Confidence intervals can be obtained by resampling: we draw Ns samples of size Nq(Z11, . . . , ZN1q), . . . , (Z1Ns, . . . , ZNNqs) from the
fitted distribution and let Z(n)1 , . . . , ZNs
(n)denote the n
t horder statistic of each sample.
A 95% confidence interval for the generalized Pareto fit is then defined as the 2.5 and 97.5 empirical percentiles of the sets {Z(n)m : m= 1,...,Ns}. When the estimator
used to obtain $θ is asymptotically normal, the uncertainty of the model parameters can be taken into account by drawing the Ns samples from different generalized
Pareto distributions whose parameters are normally distributed with means ˆξ and ˆ
σ(sl) respectively and standard deviations corresponding to the uncertainty of the
vector ˆθ; strictly positive parameters being handled on the log-scale.
(Davis and Mikosch, 2009) introduced in de Fondeville and Davison (2018), i.e., π(sl, sk)= Pr X (sl)> uq(sl)| {X (sk)> uq(sk)}∩ {r (X ) > u} , l , k= 1,...,L.
If the model is stationary and isotropic, π depends only the distance h = |sl− sk|
between the locations sland sk. The theoretical values of the model are then compared
to empirical estimates ofπ and summarized using what we call an extremogram cloud: a graphical diagnostic, which displaysπ as a function of the distance, and if relevant, the orientation of the pair (sl, sk).
Model comparison can be performed using classical likelihood criteria, such as the Akaike Information Criterion (AIC), the Composite Likelihood Information Criterion (CLIC) (Davison and Gholamrezaee, 2012) or the Continuous Ranked Probability Score (CRPS) (Gneiting and Raftery, 2007). Formal testing is possible for nested models (de Fondeville and Davison, 2018).
4.7 Discussion
Peaks-over-threshold analysis is widely used for modelling tails of univariate distri- butions through the generalized Pareto distribution, but natural hazards cannot be studied using only univariate results. In this chapter, we have extended peaks-over- threshold analysis to extremes of functional data. Exceedances are defined using a real-valued functional r , and modelled with the generalized r -Pareto process, a functional generalization of the generalized Pareto distribution, covering the three possible tail decay regimes. This family appears as the limit for r -exceedances of a properly rescaled process. We derive construction rules for generalized r -Pareto processes, give simulation algorithms and highlight their link to max-stable processes. Finally we discuss inference procedures and model validation.
The strength of the theoretical results developed in this chapter depends on the relevance of the properties satisfied by r . The class of linear functionals is particularly attractive because in this case the risk is directly defined on the original process and Corollary 4.1 gives the limit distribution of large rlin-exceedances of X . Also,
the generalized rlin-Pareto process can be factorized into two components: a radial
part measuring the intensity of the excess and an angular component modelling the dependence. This decomposition enables simulations for fixed intensities, i.e. for determined values of r (X ), and allows the generation of catalogues of extreme events for fixed return periods; such events can later be used as input for stress tests either on human infrastructure or insurance portfolios. In Chapter 5, we illustrate this methodology by developing a stochastic weather generator for extreme windstorms
over Europe.
While the class Rlinmight seems restrictive, for spatial applications it can be combined
with tools from image processing such as Fourier or wavelet transforms, that have been successfully used to classify large and complex datasets of images. This chapter opens the development of flexible, and if possible linear, risk functionals able to discriminate between different meteorological phenomena.
storms over Europe
5.1 Introduction
On 25 January 1990, the wind storm Daria, also known as the ‘Burns Day Storm’ as it started on the birthday of Scottish poet Robert Burns, struck the United Kingdom. Daria is famous for being one the most severe extra-tropical cyclones in this region. During the two days where the storm was active, 97 deaths were reported and damage was valued at around 8.2 billion dollars. The strongest gusts were measured to be 170 km.h−1, a speed equivalent to a category 1 hurricane. Figure 5.1 shows the maximum speed over the past 3h hours of the wind gusts sustained for at least 3s. The selected time steps correspond to the 24 hours during which the storm was at its peak. About 10 years later, on 26 December 1999, the storm Lothar swept across western and central Europe during a period of 36 hours. A wind speed of 169 km.h−1was recorded in Paris, and at the summit of the ‘Dole’ in Switzerland, the weather station reported a maximum wind gust of 201.2 km.h−1. Lothar was classified as a category 2 cyclone, and caused 8 billion dollars loss and more than 100 deaths.
Estimating the risk linked to such extreme natural hazards has become a major ques- tion in recent decades, because of the possible influence of global warming. Even if the influence of human activity on the climate has been established, according to the IPCC (Pachauri et al., 2014), its impact on specific types of events is much less certain. To issue long-range projections or to minimize risks linked to wind storms, both climatologists and insurers want to better understand the extremal behaviour of weather events.
In this chapter, we use the theory presented in Chapter 4 to develop a stochastic weather generator of extreme wind storms over Europe. The model can create cata- logues of wind storms with unobserved shapes and tracks, and potentially unobserved
intensities. These catalogues can then be used as ‘stress tests’ for physical infrastruc- ture or insurance portfolios.
Figure 5.1 – Maximum speed (m.s−1) over the past 3h hours of the wind gusts sustained for at least 3s from ERA-Interim reanalysis during the peak of wind storm Daria, which swept over Europe during January 1990.