We generally assume that the observations (here time series of length 37) are i.i.d , which implies that, for all t=1,…,37, the series (X1t,…,XNt) with N the number of observations should constitute an i.i.d sample. However, as illustrated by the autocorrelogram ACF and the partial autocorrelogram PACF at time t=1 (Fig. ), we observe that the time series are correlated for each value of t. The correlation level is around 0.10 when the lag in terms of tidal cycle index equals 50. Besides, we observe that the time series are cross-correlated as for instance XM1 is correlated with XM+h2 (see Appendix ).

In addition to these strong correlations, we observe a seasonality of the variable with higher values observed during Winter months (September–March) (see Appendix ). In conclusion, the independence assumption is rejected.

In this paper, we consider that time series are realizations of a stochastic process XM whose sample paths belong to some Hilbert space E . Using the notation ⟶w for weak convergence of probability measures, we say that XM is regularly varying with index α>0 if there exist normalizing constants an→∞ and some measure μ on E\{0} such that nP(an-1XM∈.)⟶wn→∞μ(.), and μ is homogeneous: μ(cA)=c−αμ(A),c>0. Note that this definition implies, by Proposition (3.1) of , that for each t, the law of XMt exhibits asymptotic Pareto-type decay with index α. To properly define this concept, we recall some prerequisites in extreme value theory. Let U be a univariate random variable and assume that U lies in the attraction domain of a so called Generalized Extreme Value (GEV) law . This means that, for a n-sample U1,…,Un following the law of U, there exists constants an>0,bn such that Mn:=max⁡i∈[[1,n]]Ui-bnan, and Mn converges in distribution to a GEV law. Then, when we consider the exceedances of U over a high threshold u – the so-called Peak-Over-Threshold (POT) approach –, we obtain a Generalized Pareto distribution (GPD): 1P(U≤x∣U>u)=1-1+γx-uσ-1γifγ≠01-exp⁡(-x-uσ)otherwise, where x>u verifies 1+γx-uσ>0. We denote this law GPD(u,σ,γ) and FGPD the corresponding cumulative distribution function (cdf), where γ,σ are respectively shape and scale parameters. Within the framework of regular variation, the law of XMt exhibits Pareto-type decay for every value of t, corresponding to the case u=σ/γ, with γ=1/α>0 in Eq. (). Following the peak-over-threshold approach , the extremeness of a time series is quantified by a homogeneous functional ℓ, which maps the time series to a scalar value. This function is referred to as a cost functional or a risk functional . A time series is classified as extreme if the corresponding value of ℓ exceeds a specified threshold. Defining the risk function as ℓ(f)=||I(f>0)f||L2, we analyze the extremes of ℓ(XM). As we estimate the evolution of γ according to the number of exceedances, we analyze the sign of the shape parameter γ (Fig. ). The Hill estimator explored in (dotted line) is based on a property of extreme observations under heavy-tail hypothesis (γ>0). If the assumption holds, the estimator converges. Since the plot shows no stable region for the Hill estimator, we conclude that the condition γ>0 is not met, which is confirmed by the moments and MLE estimators (straight and dashed line) analysed in . Hence, the regular variations assumption is not satisfied.

Starting from observed time series XM, we aim at removing the temporal dependence between successive time series. We first detrend the data by simply writing 2XMt=αtM+ct+ηMt=αtM+X̃Mt, where αt represents the slope, ct is the intercept and ηMt is the residual term. The detrended time series is denoted by X̃Mt. In practical applications, we need to subsample in the data by imposing a time step Δ between observations in order to reduce the temporal correlation, which is also known as declustering in the literature. Mathematically, we have M=1+Δm with m=1,2,…. Finally, for each value of t, we model the temporal dependence, with respect to M, by an autoregressive model, leading to 3X̃Mt=β0t+∑i=1pβitX̃M-Δit+εMt, where β0t is a constant term, (βit)i=1,…,p are the model parameters and ε1t,ε1+Δt,…,εNt are the model residuals. These residuals form a white noise, i.e. are i.i.d. random variables, that we will now consider. The parameters are estimated by applying the least-squares on the residuals. This can be viewed as a conditional log-likelihood method under a Gaussian assumption. However, apart from this estimation stage, we do not impose a specific distribution for the residuals and more specifically, no assumptions about the tail of the distribution.

We propose here an approach based on autoregressive models that does not account for seasonality. This choice is justified in the present study, as the seasonal part of XM can be easily removed by focusing on the Winter periods (see Sect. ). Nevertheless, more sophisticated models incorporating seasonal components, such as SARIMA models , could be considered. However, a limitation of these models is that they would require multiple initial time series to invert Eq. (), which forms the basis of the generation method described in Sect. .

Starting from the residuals εMt obtained in the previous step, we aim at obtaining a heavy-tailed distribution. A simple way to proceed is to use the transformation T:X↦-1/log⁡(FX(X)) where X is a continuous random variable with cdf FX. Indeed, T(X) follows a Fréchet distribution (which is heavy tailed). However, in our case, the cdf of εMt must be estimated independently for each t=1,…,37. Following , we approximate it by a mixture of the empirical cdf in the non-extreme part and a GPD distribution in the extreme region. Denoting Femp the empirical cdf and FGPD the distribution function of a GPD variable (Eq. ), F^εt has the form 4F^εt(ε)=Ftemp(ε)ifε<ut(1-pu)+puFGPD(ut,σt,γt)(ε)otherwise, where pu is a parameter and ut verifies pu=1-F^εt(ut). In the sequel, we will denote ZMt=T(εMt).

The simulation of extreme time series is based on the polar decomposition given in Eq. (). Simulating the radial component ℓ(f) is straightforward as its asymptotic distribution is explicit (a Pareto distribution). In contrast, simulating the angular component A(f) is much more challenging. Several simulation methods have been proposed in the literature , which rely on a spectral representation of the coordinate-wise maximum of ZMt using Gaussian processes . They all depend on the choice of a parametric model for the variogram. In contrast, we use a dimensionality reduction method, here PCA, before fitting parametric models. We follow by decomposing the standardized time series ΩM in an orthonormal basis where (νj)j≥1 are eigenvectors sorted by decreasing order of the eigenvalues (λj)j≥1, (Cj)j≥1 the PCA coordinates.

To reduce the dimensionality of the problem, we retain only the first J eigenvectors and denote by RJ the ratio of total inertia explained by the truncated basis. Applying the elbow rule , we truncate the PCA basis when adding another eigenvector does not significantly decrease (1-RJ). This dimensionality reduction step allows us to estimate only the joint distribution of the scores, enabling the simulation of new extreme observations.

We account for the dependence between PCA coordinates by using the copula C defined as 7F(C1,…,CJ)=C(F1(C1),…,FJ(CJ))=C(v1,…,vJ), where F is the joint distribution function of the vector (C1,…,CJ) and Fi is the distribution function of the ith marginal. We approach the joint law of (v1,…,vJ) by fitting a parametric copula model. Since many parametric copulas models are limited to the bivariate case, we use vine copulas when J>2, which use a tree representation of the dependence structure where each edge corresponds to a bivariate copula. Then, the law of the whole vector is modeled by combining the pairwise models in a hierarchical manner. Thus, this modeling approach captures multivariate distributions without requiring any assumption on the joint distribution of the entire vector v1,…,vJ. To simulate new scores Csim=(Csim,1,…,Csim,J), we apply Fi-1 for i=1,…,J to new values (ṽ1,…,ṽJ) sampled from the model. With this method, we generate nsim new angles Ωsim, which are then transformed back to the original scale by multiplying by the data standard deviation and adding the data mean at each time step. As ℓ(ΩM) must equal 1, we return Ωsim=Ωsim/ℓ(Ωsim). This simulation step makes it possible to simulate new extreme time series, which is discussed in the next section.

Thanks to the independence established in Eq. (), we generate new extreme time series Zsim by simulating new values of the radius ℓ(f) and by combining them with the simulations Ωsim. As ZMt∼Fréchet(1), we impose that Zsimt>0 for every value of t. To this aim, we apply a rejection sampling: if a curve is strictly above 0, we accept it; otherwise, we discard it and we simulate both a new Zsim and a new radius. The final step consists in applying the reversed transformation T-1 to the accepted time series, yielding extreme time series εsim, which can be used to simulate new realizations of the time series X̃M verifying ℓ(ZM)>uℓ.

In the case where p=1, we obtain simulated time series for (X̃simt)t=1,…,37, depending on initial time series (X̃M-Δt)t=1,…,37 with 8X̃simt=β0t+β1tX̃M-Δt+εsimt.

The values and the shape of X̃sim depend on the levels reached by X̃M-Δ. In this context, we can use an extreme X̃M-Δ to produce consecutive extremes. As our objective is to simulate data-like extreme time series, we must choose an appropriate time series X̃M-Δ given εsim. Thus, we can adopt a conditional or an unconditional approach to sample X̃sim. If we choose a conditional sampling, we account for the joint distribution of the couple (X̃M-Δt,εMt) as these variables may still exhibit dependence, which should be preserved during the sampling process. A simple approach to incorporate this dependence is to sample from the empirical distribution of (ℓ(X̃M-Δ)∣ℓ(εM)=x). For each value x of ℓ(εsim), we define a symmetric window centered at x that include 20 neighboring data points. Then, we draw randomly with replacement one point from this window. Finally, following Eq. (), we incorporate the trend of XMt over the period by using the equation 9Xsim,Mt=αtM+X̃simt, where M is known thanks to the sampling of the X̃M-Δ. Another approach would be to predict today's forcing conditions by using the current index. Yet, we focus on the detrended time series to facilitate direct comparisons between our simulations and the observed extreme time series. We apply our method in the next section to the case study and discuss the validation of the simulator.

First, due to the seasonal behavior identified in Sect. , we restrict our analysis to Winter surge time series. In this context, we redefine XM as the Mth Winter surge time series.

Then, by applying the whitening stage described in Sect. , we detrend the time series XMt with Eq. () and consider the detrended time series X̃Mt. Since significant correlations remain when lag h>1 (see PACF in Fig. ), we retain for the sake of simplicity only one observation out of 3, corresponding to Δ=3. The following graphic (Fig. ) illustrates the database construction, where the grey rectangles represent the tidal cycles that are excluded from the analysis.

We now analyze the pair of correlated variables (X̃Mt,X̃M+Δht). The autocorrelations and partial autocorrelations (respectively Fig. a and b) are not significant for a lag h≥2, which confirms the validity of our hypothesis for an AR(1) model. This is confirmed by the ACF and the PACF of the residuals (see Fig. ), which show no significant temporal correlations.

We obtain also non significant cross-correlations for the joint vectors ((εMt)t=1,…,37,(εM+Δht)t=1,…,37), which shows that we comply with the framework assumptions highlighted in Sect. . Please refer to Appendix  for more details. Besides, since we study εMt's exceedances for a given t, we assess extremal dependence as defined in . To this end, we apply the statistical test proposed by whose null hypothesis is the asymptotic dependence of the residuals εMt. The null hypothesis is rejected at confidence level 1% (p value < 10⁻¹⁰), indicating that the residuals εMt are asymptotically independent. Then, as detailed in Sect. , we aim at obtaining heavy-tailed marginals for each value of t. To achieve this, following , we estimate the cdf of εMt and apply the transformation T to the residuals, obtaining Fréchet marginals ZMt for each value of t. An intermediate step involves selecting the threshold parameter ut used in Eq. (). To guide this choice, we use the property that for any (wt>ut) the exceedances (εMt>wt) should follow a GPD law if ut is a suitable threshold. Accordingly, we define updated parameters (σ′,γ′) that remain stable as wt increases. With the same objective, we analyze, for each value of t, the behaviour of the mean residual life (MRL) E(εMt-ut∣εMt>ut) as a function of ut. Guided by these tools, we define ut as the 90% percentile of εMt since the MRL plot becomes linear beyond this point and the couple (σ′,γ′) remains stable for all thresholds (wt>ut) (see Appendix ). To assess the quality of the marginal transformation, we compare empirical quantiles ZMt with those of the Fréchet distribution across various values of t. The levels obtained are close to the theoretical ones, demonstrating the consistency of the transformation. Having marginals ZMt with Pareto-type tails is a necessary but not sufficient condition for ZM to be a regularly varying random element of L2([0,1]), as the latter additionally requires a joint tail condition (see Sect. ). As a consequence, we now show that ZM is regularly varying in the L2 sense with index α=1. By Theorem (3.1) of , ZM is a regularly varying random element with index α if and only if ΩM converges in distribution and the random variable ℓ(ZM) has a Pareto-type tail with index α. Here, we can be confident that both conditions are satisfied. First, the analysis of the mean projection of ΩM supports the convergence in law of the angular component (see Appendix ).

Secondly, using Eq. (), the transformation induces a change in the tail behavior of the distribution of ℓ(.): while ℓ(εM) is light-tailed distributed (see Fig. a), ℓ(ZM) displays a Pareto-type tail with index α=1/γ≈1 (Fig. b), as confirmed by the convergence of the Hill estimator.

As we get back to the framework assumptions with the pre-processing step, we now choose the threshold uℓ that defines the set of extreme observations via the condition ℓ(ZM)>uℓ of Eq. (). Following Sect. , the choice of uℓ depends on the convergence behavior of ΩM when uℓ goes to infinity. In our case, using the argument detailed in Sect. , ΩM appears to converge in distribution when the number of extreme observations is between 200 and 300, corresponding to approximately 4% and 6% of the database. As a consequence, we select the top 5% of the dataset, corresponding to approximately 7 extreme observations per year, and represent a sample of the resulting n=259 time series on their original scale and on the Fréchet scale (Fig. ). The figure shows that applying the marginal transformation amplifies the variations between successive measures.

We use the polar decomposition defined in Eq. () as a basis for our simulation method. After selecting the n=259 extreme observations, we first compute their angular component ΩM and derive the corresponding PCA basis. Following Sect. , we select the first J=3 eigenvectors of the basis as the ratio (1-RJ) slowly decreases beyond the third dimension (see Appendix C: Fig. ). With this choice, the selected dimensions explain RJ=83% of the variance. We use vine copulas and model the joint distribution of the coordinate vector by fitting a range of copula families, namely elliptical and Archimedean copulas such as Tawn and Clayton models, to the bivariate vectors defined by the vine structure. Finally, for each pair of coordinates, we use the Akaike Information Criterion (AIC) to select the best copula and estimate the tail coefficients (λ+,λ-) along with the Kendall's τ. In our case, a t copula is selected for each pair of coordinates. Please refer to Appendix  for additional information about the modeling. The iso-density contours of the fitted law are consistent with the distribution of the data points. To go further, as we assume that the observations (v1,…,vJ) follow the fitted copula model, we perform some goodness-of-fit tests presented in . We do not reject the null hypothesis when we use the bootstrap test based on the White statistic (p value ≈0.5). However, following , using Student copula implies that there is an asymptotic dependence between coordinates. Figure  shows that this assumption does not hold for the pair (C1,C2). In this case, we can address this issue for the first two coordinates by adopting the second best model, namely the rotated Tawn 1 copula , which is well suited for modeling asymptotic independence. In contrast, since the pair (C2,C3) exhibits asymptotic dependence, the assumption of the Student copula model remains valid. Using the composite copula model, we simulate new vectors (vsim,1,vsim,2,vsim,3) and transform each coordinate back to its initial scale by applying Fi-1 for i=1,…,3. We simulate nsim=2000 triplets of coordinates from the model and we compare the simulated values with the coordinates of the extreme observations (Fig. ). The marginals obtained in the simulations are consistent with the data distributions, indicating that our model is consistent with the dependence structure of the coordinate vector.

We use the coordinates Csim to obtain the angular components Ωsim. The resulting patterns resemble that of the extreme observations although the simulated curves appear to be smoother.

Following Sect. , we generate extreme time series Zsim by combining the simulated angles Ωsim with simulations of ℓ(ZM). Then, we apply the inverse transformation T-1 to Zsim to return to the original scale, which produces simulated extreme residuals εsim. Using Sect. , we now invert the estimated AR(1) models to return simulated time series X̃sim. We consider two choices of X̃M-Δ: one observed during the extreme storm Johanna (10 March 2008) and another with a non-extreme behavior (15 February 1986), verifying ℓ(T(εM))<uℓ. The results, displayed in Fig. , show that the choice of X̃M-Δ significantly affects both the values and the shape of X̃sim. The left panel reveals notably high values, illustrating the effect of having consecutive extremes.

Since the variables (X̃M-Δt,εMt) are dependent, we apply the conditional sampling illustrated in Sect.  (see Appendix ). Then, we generate new time series X̃sim by using εsim and the selected X̃M-Δ (see Fig. ). We see that the shape and the values of simulated and observed extreme time series are quite similar, suggesting that our simulations are consistent with the observations. However, we can use other methods to validate our simulator, which is discussed in detail in the following.

First, we compare the empirical percentiles at each time point t with those derived from the simulated time series (Fig. ). To quantify uncertainty in the observations, we construct a bootstrap 95% confidence band by resampling the extreme time series 500 times. The simulated levels obtained are coherent with the observations as the simulations' percentiles lie within the data confidence band.

The simulated extreme time series should exhibit shape similarities with the observed extremes. One simple way to compare the shape of the time series is to use dimensionality reduction methods and to compare the resulting coordinates. To this end, we analyse the angles of the time series as the simulations extrapolate the L2 norm of the observations. We apply PCA to the time series X̃/ℓ(X̃) and, to preserve potential differences, both simulated and observed series are further normalized by using the mean and the standard deviation of the observations. Then, we compare the PCA coordinates of recorded time series with those of simulated time series (Fig. ). The similarity in the score distributions suggests that simulated time series and extreme observations exhibit comparable behavior. Yet, this similarity is more pronounced in the first dimension, which accounts for the largest proportion of variability (here 60%). A KS test confirms that simulations' and observations' coordinates follow the same law in the first dimension (p value ≈27%) but do not in the second dimension (p value ≈7.7×10-5).

Then, we compare the temporal dependence within each tidal cycle between observations and simulations, with a focus on extremal dependence. Following , we assume that X̃Mt and X̃Ms are asymptotically dependent for all pairs (t,s). To quantify the dependence, we estimate the ℓ-extremogram for both the recorded and simulated time series. We detail the approach in Appendix . We focus on the values exceeding the 90% percentile and use 500 resamples to construct a bootstrap 95% confidence band. The extremogram of the simulated time series lies within the confidence band (Fig. ) based on the observed extreme time series.

Moreover, the density obtained at each time t should be an extrapolation of the observed data. Directly using the quantiles of the simulations may produce very large values. This occurs because we simulate X̃sim verifying ℓ(T(εM))>uℓ, meaning their extremes can be larger than those observed. In this setting, we compare our values with the empirical quantiles of the observations and the fitted law of exceedances. Following , the results are expressed in terms of return period P and corresponding return levels, with approximately 7 extreme observations per year (Fig. ). All implementation details are provided in Appendix . The highest simulated levels fall within the confidence band, indicating that our simulations look similar to the extreme observations. Thus, they can be considered a reliable extrapolation of the observed data toward higher values.

Finally, following the same idea as two-sample classification tests , we rely on a classification-based approach to determine whether the generated extreme time series differ from the original dataset. We define a binary classification problem, where the target variable y equals 1 if the input x is a simulated time series, 0 otherwise. The goal is to learn a function g that, given the input x, predicts the correct class y. If the simulations behave like the extreme observations, such classifier should hardly identify the simulated time series, resulting in an accuracy close to 50%. To address this question, we begin by randomly drawing n=259 simulated time series without replacement (Step 1) since the number of simulated time series nsim=2000 differs from n. We construct a dataset composed of these sampled time series and extreme observations, ensuring that simulated time series make up 50% of the data. Then, after splitting this dataset into training and testing subsets, we train our model on the training part (Step 2). Finally, we evaluate the model's overall accuracy on the test set (Step 3). To account for the influence of the sampling used in Step 1, we repeat 100 times these three steps and provide a confidence interval of order 90% for the overall accuracy. We use three classifiers with this approach: a support-vector machine (SVM) with a radial kernel, a generalized linear model (GLM), and a random forest . We apply this procedure with different types of input features: the time series, its norm and its angle. If we provide the time series, all classifiers achieve an overall accuracy close to 50% (see Table ), meaning that the ML models struggle to identify simulated time series. This suggests that our simulations are quite consistent with the observed data. For example, the distributions of ℓ(X̃sim) and ℓ(X̃M) are quite similar (see Appendix ), with differences appearing for the largest values as the simulations better explore the extremes of ℓ(X̃M). In contrast, when using the angle as input feature, the models are better able to distinguish between simulations and observations but the accuracy still remains low on the order of 60 % in average.

This approach also reveals the influence of X̃M-Δ as the attained accuracy depends on the sampling method, which is discussed in Appendix . The classifiers identify more easily the simulated time series when an unconditional sampling is used. For instance, the overall accuracy exceeds 50 % when the time series itself is used as input.