

\section{Introduction}

As stated in Chapters~\ref{chap:challenges} and \ref{chap:acer}, extreme value statistics, even in applications, is generally based on asymptotic results. This is done either by assuming
that the epochal extremes, for example yearly extreme wind speeds at a given location,
are distributed according to the generalized (asymptotic) extreme
value distribution with unknown parameters to be estimated on the
basis of the observed data, cf. Chapter~\ref{chap:aevt}. Or it is assumed that the exceedances
above high thresholds follow a generalized (asymptotic) Pareto
distribution with parameters that are estimated from the data, cf. Chapter~\ref{chap:pot}. 
With the ACER method now available, the performance of these three methods on simulated or measured data may be compared. Note that all calculations of the empirical ACER functions in this chapter were performed using the ACER program package for Matlab \citep{ACERmanual}\index{ACERmanual}. This package also allows for optimized fitting of parametric functions for prediction of long return period extreme values for the case of asymptotic Gumbel distributions, cf. Section~\ref{P}, which totally dominates engineering applications.

The first example in this chapter deals with synthetic data, which allows us to control the exact result to be predicted by the different methods. The second example looks at measured wind speed data at three locations along the coast of Norway. The measurement periods are ranging from 12 to 16 years. The goal would then e.g. be to predict a 100 year return period value for the wind speed on the basis of these data. This represents a classical problem in wind engineering. For some other problems of these kind of engineering challenges where the ACER method has been applied, cf. e.g. \cite{gastna16jsr,gastna18ms,yuwuxiwana20aor,sinalech22coe} 


\section{Extreme Value Prediction for Synthetic Data}\label{sec:SD}\index{extreme value prediction!synthetic data}

In this section the performance of the ACER method and also the 95\% CI estimation will be illustrated. 
20 years of synthetic wind speed data are subjected to analysis, amounting to 2000 data points, which is not much for detailed statistics. However, this case may represent a real
situation when nothing but a limited data sample is available. In
this case it would appear crucial to provide extreme value estimates
utilizing all data available. As will be demonstrated, the tail
extrapolation technique proposed for the ACER method performs on the average better than the asymptotic POT or Gumbel methods.

The extreme value statistics will be analyzed by application
to synthetic data for which the exact extreme
value statistics can be calculated \citep{nacl01}. In particular, it is assumed that the
underlying (normalized) stochastic process $Z(t)$ is stationary
and Gaussian with mean value zero and standard deviation equal to
one. It is also assumed that the mean zero up-crossing rate
$\nu^+(0)$ is such that the product $\nu^+(0)T = 10^3$ where
$T=1$~year, which seems to be typical for the wind speed process.
Using the Poisson assumption, the distribution of the yearly
extreme value of $Z(t)$ is then calculated by the formula
\begin{equation}\label{eq:1yr}
F^\textrm{1yr}(\eta) = \exp\left\{-\nu^+(\eta)T\right\} =
\exp\left\{-10^3\exp\left(-\frac{\eta^2}{2}\right)\right\} \,,
\end{equation}
where $T=1$~year and $\nu^+(\eta)$ is the mean up-crossing rate
per year, $\eta$ is the scaled wind speed. The 100-year return
period value $\eta^\textrm{100yr}$ is then calculated from the
relation $F^\textrm{1yr}(\eta^\textrm{100yr})=1-1/100$, which
gives $\eta^\textrm{100yr}=4.80$.


The Monte Carlo simulated data to be used for the synthetic
example are generated based on the observation that the peak
events extracted from measurements of the wind speed process, are
usually separated by 3-4 days. This is done to obtain
approximately independent data, as required by the POT method. In accordance
with this, peak event data are generated from the extreme value
distribution
\begin{equation}\label{eq:3d}
F^\textrm{3d}(\eta) = \exp\left\{- q
\exp\left(-\frac{\eta^2}{2}\right)\right\} \,,
\end{equation}
where $q=\nu^+(0)T=10$, which corresponds to $T=3.65$~days, and $F^\textrm{1yr}(\eta) = \big( F^\textrm{3d}(\eta) \big)^{100}$.


Since the data points (i.e. $T=3.65$~days maxima) are independent,
${\varepsilon}_k(\eta)$ is independent of $k$. Therefore,
$k = 1$ is chosen. Since there are 100 data from one year,
the data amounts to 2000 data points. For estimation of a 95\% confidence interval for each value of the ACER function ${\varepsilon}_1(\eta)$ for the chosen range of $\eta$-values, the required standard deviation in Eq.~\eqref{eq:cb} was based on 20 estimates of the ACER function using the yearly data. This provided a 95\% confidence band on the ACER function based on 2000 data. From these data, the predicted 100~year return level is obtained from $\hat{\varepsilon}_1({\eta}^\textrm{100yr}) = 10^{-4}$. A nonparametric bootstrapping method was also used to estimate a 95\% confidence interval based on 1000 resamples of size 2000.


The POT prediction of the 100~year return level was based on using maximum likelihood estimates (MLE) of the parameters in Eq.~\eqref{eq:pot.4} for a specific choice of threshold. The 95\% confidence interval was obtained from the parametrically bootstrapped density of the POT estimate for the given threshold. A sample of 1000 data sets was used. One of the unfortunate features of the POT method is that the estimated 100~year value may vary significantly with the choice of threshold. So also for the synthetic data. We have followed the standard recommended procedures for identifying a suitable threshold \citep{cole01}.

Note that in spite of the fact that the true asymptotic distribution of exceedances is the exponential distribution Eq.~\eqref{eq:pot.5}, the POT method used here is based on adopting Eq.~\eqref{eq:pot.4}. The reason is simply that this is the recommended procedure \citep{cole01}, which is somewhat unfortunate, but understandable. The reason being that the GP distribution provides greater flexibility in terms of curve fitting. If the correct asymptotic distribution of exceedances had been used on this example, poor results for the estimated return period values would be obtained. The price to pay for using the GP distribution, is that the estimated parameters may easily lead to an asymptotically inconsistent distribution.    

The 100~year return level predicted by the Gumbel method was based on using the method of moments for parameter estimation on the sample of 20 yearly extremes. This choice of estimation method is due to the small sample of extreme values. The 95\% confidence interval was obtained from the parametrically bootstrapped density of the Gumbel prediction. This was based on a sample of size 10,000 data sets of 20 yearly extremes. The results obtained by the method of moments were compared with the corresponding results obtained by using the maximum likelihood method. While there were individual differences, the overall picture was one of very good agreement. 


In order to get an idea about the performance of the ACER, POT and Gumbel methods, 100
independent 20 yr MC simulations as discussed above were done. Table~{\ref{tab:T1}}
compares predicted values and confidence intervals for a selection of 10 cases together with average values over the 100 simulated cases. It is seen that the average of the 100 predicted 100~year return levels is slightly better for the ACER method than for both the POT and the Gumbel methods. But more significantly, the range of predicted 100~year return levels by the ACER method is 4.34 - 5.36, while the same for the POT method is 4.19 - 5.87, and for the Gumbel method 4.41 - 5.71. Hence, in this case the ACER method performs consistently better than both these methods. It is also observed from the estimated 95\% confidence intervals that the ACER method, as implemented in this book, provides slightly higher accuracy than the other two methods. Lastly, it is pointed out that the confidence intervals of the 100~year return level values by the ACER method obtained by either the simplified extrapolated confidence band approach or by nonparametric bootstrapping are very similar except for a slight mean shift. As a final comparison, the 100 bootstrapped confidence intervals obtained for the ACER and Gumbel methods missed the target value three times, while for the POT method this number was 18.


\begin{table}[!htb]
\begin{center}
\caption{100 year return level estimates and 95\% CI (BCI = CI by bootstrap) for A=ACER, G=Gumbel and P=POT. Exact value = 4.80. 
\label{tab:T1}}
\begin{tabular}{c c c c c c c c}
\hline
$Sim. No.$ & $\textrm{A}$ $\hat{\eta}^{100yr}$ & $\textrm{A CI}$ & $\textrm{A BCI}$ & $\textrm{G}$ $\hat{\eta}^{100yr}$ & $\textrm{G BCI}$ & $\textrm{P}$ $\hat{\eta}^{100yr}$ & $\textrm{P BCI}$ \\
\hline
 1  & 5.07 & (4.67, 5.21) & (4.69, 5.42) & 4.41 & (4.14, 4.73) & 4.29 & (4.13, 4.52) \\
10  & 4.65 & (4.27, 4.94) & (4.37, 5.03) & 4.92 & (4.40, 5.58) & 4.88 & (4.42, 5.40) \\
20  & 4.86 & (4.49, 5.06) & (4.44, 5.19) & 5.04 & (4.54, 5.63) & 5.04 & (4.48, 5.74) \\
30  & 4.75 & (4.22, 5.01) & (4.33, 5.02) & 4.75 & (4.27, 5.32) & 4.69 & (4.24, 5.26) \\ 
40  & 4.54 & (4.20, 4.74) & (4.27, 4.88) & 4.80 & (4.31, 5.39) & 4.73 & (4.19, 5.31) \\
50  & 4.80 & (4.35, 5.05) & (4.42, 5.14) & 4.91 & (4.41, 5.50) & 4.79 & (4.31, 5.34) \\
60  & 4.84 & (4.36, 5.20) & (4.48, 5.19) & 4.85 & (4.36, 5.43) & 4.71 & (4.32, 5.23) \\
70  & 5.02 & (4.47, 5.31) & (4.62, 5.36) & 4.96 & (4.47, 5.53) & 4.97 & (4.47, 5.71) \\
80  & 4.59 & (4.33, 4.81) & (4.38, 4.98) & 4.76 & (4.31, 5.31) & 4.68 & (4.15, 5.27) \\
90  & 4.84 & (4.49, 5.11) & (4.60, 5.30) & 4.77 & (4.34, 5.32) & 4.41 & (4.23, 4.64) \\
100 & 4.62 & (4.29, 5.05) & (4.45, 5.09) & 4.79 & (4.31, 5.41) & 4.53 & (4.05, 4.88) \\
\hline
\textrm{Av. 100} & 4.82 &  (4.41, 5.09) & (4.48, 5.18) & 4.84 & (4.37, 5.40) & 4.72 & (4.27, 5.23)  \\
\hline
\end{tabular}
\end{center}
\end{table}


An example of the ACER plot and results obtained for one set of data is presented in Figure~\ref{fig:S25}. The predicted 100 year value is 4.85 with a predicted 95\% confidence interval (4.45,~5.09). Figure~\ref{fig:S25POT} presents POT predictions based on MLE for different thresholds in terms of the number $n$ of data points above the threshold. The predicted value is 4.7 at $n = 204$, while the 95\% confidence interval is (4.25,~5.28).  The same data set as in Figure~\ref{fig:S25} was used.  This was also used for the Gumbel plot shown in Figure~\ref{fig:gumbel}. In this case the predicted value based on the method of moments (MM) is $\hat{\eta}_\textrm{MM}^\textrm{100yr}$ = 4.75 with a parametric bootstrapped 95\% confidence interval of (4.34,~5.27).  
Prediction based on the Gumbel-Lieblein BLUE method (GL), cf. e.g. \cite{cook85}, is $\hat{\eta}_\textrm{GL}^\textrm{100yr}$ = 4.73
with a parametric bootstrapped 95\% confidence interval equal to (4.35,~5.14).



\begin{figure}[!hbt]
\begin{center}
\dofig{7cm}{Figures_jps/ACER_1_fit.pdf}%\center
\end{center}
\caption{Synthetic data ACER $\hat{\varepsilon}_1$, Monte Carlo
simulation (*); optimized curve fit (---); empirical 95\%
confidence band (-~-); optimized confidence band ($\cdot \cdot
\cdot \cdot$). Tail marker $\eta_1 = 2.3$ } \label{fig:S25} 
\end{figure}



\begin{figure}[!hbt]
\begin{center}
\dofig{7cm}{Figures_jps/POT_1_fit.pdf}%\center
\end{center}
\caption{The point estimate $\hat{\eta}^\textrm{100yr}$ of the
100-year return period value based on 20 years synthetic data as a
function of the number $n$ of data points above the threshold. The return
level estimate = 4.7 at $n=204$.} \label{fig:S25POT} 
\end{figure}



\begin{figure}[!hbt]
\begin{center}
\dofig{7cm}{Figures_jps/EV_1_fit_last.pdf}%\center
\end{center}
\caption{The point estimate $\hat{\eta}^\textrm{100yr}$ of the
100-year return period value based on 20 years synthetic data. 
Lines are fitted by the method of moments -- solid line (---) 
and the Gumbel-Lieblein BLUE method -- dash-dotted lite (--\,$\cdot$\,--). The return
level estimate by the method of moments is 4.75, by the Gumbel-Lieblein BLUE method is 4.73} \label{fig:gumbel} 
\end{figure}






\section{Measured Wind Speed Data}\label{M1}\index{extreme value prediction!measured wind speed data}

In this section we analyze real wind speed data, measured at three weather stations off the coast of Norway: At Torsv{\aa}g Fyr weather station (station number 90800)\index{wind speed measurements!Torsv{\aa}g Fyr}, 
Sula weather station (station number 65940)\index{wind speed measurements!Sula}
and Obrestad Fyr weather station (station number 44080)\index{wind speed measurements!Obrestad Fyr}, cf. \cite{kana13jweia}.
Figure~\ref{figP02} shows the geographical position of each station. The
hourly maximum of the three seconds wind gust (10 meters above the ground) 
were recorded during 13 years (1997-2010) at the first station, 12 years (1998-2010) at the second, and
16 years (1994-2010) at the third station \citep{eklima}. 

%(Maximum gust (3 seconds; last hour; 10 meter above ground))
%\textit{Kraftigste vindkast (3 sekunder; siste time; 10 meter over bakken)}
%%
\begin{figure}[h!]
	\centerline{\includegraphics[scale=0.12]{JWEIA_Figures/02_Norge.pdf}}
	\caption{Map of Norway with marked weather stations}
	\label{figP02}
\end{figure}
%%

Extreme wind speed prediction is an important
issue for design of structures exposed to the weather variations.
Significant efforts have been devoted to the problem of predicting
extreme wind speeds on the basis of measured data by various
authors over several decades, see e.g.
\cite{cook82,naes98jem,pabrliad99,perrin} for extensive references
to previous work.


The objective is to estimate a 100 year wind speed for each of these locations. 
Variation in the wind speed caused by seasonal 
variations in the wind climate during the year makes the wind
speed a non-stationary process on the scale of months. Moreover,
due to global climate change, yearly statistics may vary on the
scale of years. The latter is, however, a slow process and for the
purpose of long-term prediction, it is assumed here that within a time
span of 100 years, a quasi-stationary model of the wind speeds
applies. This may not be entirely true, of course, but for the three stations under study, no apparent trend in the wind speed was detected over the period of registration.


Figures~\ref{figP03}~-~\ref{figP05} demonstrate the plots of the time series observed from each station. A conspicuous feature of the time series is the clear seasonal variation of the wind speeds. Note the paucity of data at certain times at the Sula station. The practical consequence of this is to shorten the effective length of the time series. It is of some importance to note that the samples from Torsv{\aa}g Fyr and Obrestad Fyr stations contain outlying observations, such as 45.3 m/s in June 06, 1997; 43.7 m/s in May 10, 2001 and 60.8 m/s in September 09, 2008 for Obrestad Fyr station
and 45.3 m/s in July 12, 1998 and July 31, 1999 for Torsv{\aa}g Fyr station. Such wind speeds are most likely spurious for the corresponding time periods and latitudes. Moreover, observations from the weather stations in the close neighborhood of Obrestad Fyr confirm that no heavy storm has occurred during the period in question, while no information from the stations in the neighborhood of 
Torsv{\aa}g Fyr is available. Therefore, the outliers from Obrestad Fyr station have to be rejected, while the outliers from Torsv{\aa}g Fyr are kept, mainly in order to show that the ACER method is largely insensitive to observations of this kind.


\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/03_Torsvag_TS.pdf}
	\caption{Observations from Torsv{\aa}g Fyr station}
	\label{figP03}
\end{figure}
%%
\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/04_Sula_TS.pdf}
	\caption{Observations from Sula station}
	\label{figP04}
\end{figure}
%%
\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/05_Obrestad_TS.pdf}
	\caption{Observations from Obrestad Fyr station}
	\label{figP05}
\end{figure}
%%


In Figures~\ref{figP06}~-~\ref{figP08}, $\hat{{\varepsilon }}_k (\eta )$ is plotted versus 
wind speeds, $\eta/\sigma$ for different values of $k$ for the three stations. The figures reveal 
that there is significant dependence between the data, but that this is 
largely accounted for by $k = 48$ since
there is a marked degree of convergence 
of the $\hat{{\varepsilon }}_k (\eta )$ for $k \ge 48$. 
It should be noted that $k = 48$ obviously corresponds to two days separated exceedances for hourly observations.
For $k \ge 96$, that is four days declustered data, full 
convergence has been achieved for all practical purposes. However, Figures~\ref{figP06}~-~\ref{figP08} 
also reveal that for extreme value estimation $\hat{{\varepsilon }}_1(\eta )$ 
can be used since the ACER functions all converge in the far tail. This 
clearly demonstrates the power of an ACER function plot as a diagnostic tool to 
decide on the value of $k$ needed for extreme value estimation in a particular case. In spite of 
significant dependence effects for the lower wind speeds, for the extreme 
wind speeds this is largely absent. This makes it possible to choose $k = 1$, 
which makes much more data available for estimation, with a possible 
reduction of uncertainty in estimation as a result.
%%
\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/06_TORSVAG_all_ACER.pdf}
	\caption{ACER estimates for different degrees of conditioning. Torsv{\aa}g Fyr wind gust statistics based on 13 years of hourly data with outliers included, $\sigma = 5.30$.}
	\label{figP06}  
\end{figure}    
%%
\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/07_SULA_all_ACER.pdf}
	\caption{ACER estimates for different degrees of conditioning. Sula wind gust statistics based on 12 years of hourly data, $\sigma = 5.49$.}
	\label{figP07}  
\end{figure}    
%%
\begin{figure}[!htp]
	\centering
	\includegraphics[width=2.8in]{JWEIA_Figures/08_Obrestad_all_ACER.pdf}
	\caption{ACER estimates for different degrees of conditioning. Obrestad Fyr wind gust statistics based on 16 years of hourly data, $\sigma = 5.47$.}
	\label{figP08}  
\end{figure}    
%%


Figures~\ref{figP09}~-~\ref{figP11} show the plots of the optimized fit to the 
data for $\hat{{\varepsilon }}_1 (\eta )$ for each station. The 100-year return level value 
${\eta}^{100yr}$ and its 95\% CI are estimated parametrically. For the data with outliers from Torsv{\aa}g Fyr weather station,  
$\hat{\eta}^{100yr} = 47.46$ m/s and the 95\% CI = (42.11, 50.71) with parameters of the optimal curve: $q$ = 0.44,  $b$ = 9.02, $a$ = 0.1,  $c$ = 1.33.
The predicted 100-year return wind speed and 95\% CI for the data without outliers are $\hat{\eta}^{100yr} = 47.21$ m/s; CI = (39.94, 50.60) with 
optimal parameters $q$ = 0.47, $b$ = 8.49, $a$ = 0.09, $c$ = 1.36. In the case of the Sula wind station, $\hat{\eta}^{100yr} = 46.33$ m/s; CI = (43.41, 47.77),
where the parameters of the optimal curve are: $q$ = 0.58, $b$ = 0, $a$ = 0.005, $c$ = 2.07. Finally, for the Obrestad Fyr station, the 100-year return level value is $\hat{\eta}^{100yr} =48.38$ m/s
with confidence interval (43.18, 50.74) and optimal parameters $q$ = 0.29, $b$ = 12.34, $a$ = 0.13, $c$ = 1.27.
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/09_TORSVAG_ACER1.pdf}}
	\caption{Plot for Torsv{\aa}g Fyr of $\hat{{\varepsilon }}_1 (\eta)$ against $\eta/\sigma$ on a logarithmic scale with $\eta_{1} = 2.07\sigma$ for the optimized parameter values with the 95{\%} confidence band, $\sigma = 5.30$.}
	\label{figP09}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/10_SULA_ACER1.pdf}}
	\caption{Plot for Sula of $\hat{{\varepsilon }}_1 (\eta )$ against $\eta/\sigma$ on a logarithmic scale with $\eta_{1} = 2.36\sigma$ for the optimized parameter values with the 95{\%} confidence band, $\sigma = 5.49$.}
	\label{figP10}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/11_Obrestad_ACER1.pdf}}
	\caption{Plot for Obrestad Fyr of $\hat{{\varepsilon }}_1 (\eta )$ against $\eta/\sigma$ on a logarithmic scale with $\eta_{1} = 2.5\sigma$ for the optimized parameter values with the 95{\%} confidence band, $\sigma = 5.47$.}
	\label{figP11}
\end{figure}
%%



The annual maxima method is applied to the wind gust data to compare the estimated 100-year return level values. The application of this method to the wind data is based on the premise that the time series of the yearly maxima can be modelled as a sequence of iid random variables, which would seem to be a reasonable assumption to make as a first approximation.
The Gumbel estimate $\hat{\eta}^{100yr}$ is based on the method of moments (MM) and the Gumbel-Lieblein BLUE method (GL)\index{Gumbel distribution!Gumbel-Lieblein BLUE method}, cf. e.g. \cite{cook85}.
A computer program has been written in the Matlab language to implement both methods.
Figures~\ref{figP12}~-~\ref{figP14} present observed yearly extremes extracted 
from the hourly data together with fitted straight lines on Gumbel probability paper. 
Hereby, the 100-year return level values for the first station with outliers included are $\hat{\eta}_{MM}^{100yr} = 51.33$ m/s and $\hat{\eta}_{GL}^{100yr} = 51.57$ m/s, while in the case of
rejected outlying observations, $\hat{\eta}_{MM}^{100yr} = 44.31$ m/s and $\hat{\eta}_{GL}^{100yr} = 45.84$ m/s. For Sula and Obrestad Fyr stations, the 100-year return level values are $\hat{\eta}_{MM}^{100yr} = 48.66$ m/s with $\hat{\eta}_{GL}^{100yr} = 52.9$ m/s and $\hat{\eta}_{MM}^{100yr} = 48.59$ m/s with $\hat{\eta}_{GL}^{100yr} = 53.79$ m/s, respectively.

Despite the fact that the Gumbel-Lieblein BLUE method is considered as one of the best available conventional Gumbel methods, the application of the GL method requires tables of the BLUE coefficients which are not easily available for annual data with sample size $N>24$.
Observed results reveal sensitivity of this method to outliers, which also applies for the method of moments.
It is also noted that the Gumbel-Lieblein BLUE method seems to have a tendency to overestimate predicted return level values, while the method of moments seems to be reasonably stable for the studied sets of data.


\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/12_TORSVAG_GUMBEL.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value by the Annual maxima method. Lines are fitted by the method of moments -- solid line (---) and the Gumbel-Lieblein BLUE method -- dash-dotted lite (--\,$\cdot$\,--). Torsv{\aa}g Fyr wind speed statistics, 13 years of hourly data, $\sigma = 5.30$.}
	\label{figP12}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/13_SULA_GUMBEL.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value by the Annual maxima method. Lines are fitted by the method of moments -- solid line (---) and the Gumbel-Lieblein BLUE method -- dash-dotted lite (--\,$\cdot$\,--). Sula wind speed statistics, 12 years of hourly data, $\sigma = 5.49$}
	\label{figP13}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/14_Obrestad_GUMBEL.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value by the Annual maxima method. Lines are fitted by the method of moments -- solid line (---) and the Gumbel-Lieblein BLUE method -- dash-dotted lite (--\,$\cdot$\,--). Obrestad Fyr wind speed statistics, 16 years of hourly data, $\sigma = 5.47$}
	\label{figP14}
\end{figure}
%%




The POT method is also applied to the wind gust time series. Immediately, this would seem to be an unwarranted approach since the time series of wind speeds is conspicuously nonstationary. Efforts have been made to account for the seasonal variations by explicit modelling of the parameters of the GP distribution, cf. \cite{cole01}. However, this does not seem to be a widely adopted procedure. Instead, the POT method is applied directly to the full time series, recognizing that the extracted relevant data comes from a period of 3 or 4 months, which may be considered to represent a more or less stationary period. And by manipulating time frames, this is considered to represent one year. This trick approximately circumvents the problem of nonstationarity. This will be our approach here. 

Following the \cite{WAFO-group2000Wafo}, data were declustered beforehand.
It was done in such a way that peak events separated by 3.5 days or more were extracted from the measured data 
and selected for the analysis to achieve approximate independence of the exceedances \citep{AN98b}.
Figures~\ref{figP15}~-~\ref{figP17} present POT estimates of $\eta^{100yr}$ for different threshold numbers based on maximum likelihood estimation, cf. Chapter~\ref{chap:pot}. Estimates were obtained by using the \cite{MATLAB2009} Statistics Toolbox routine \textsf{gpfit}. It is interesting to observe the unstable characteristics of the estimates over a range of threshold values. While it is clearly of interest to discuss methods for stabilizing the POT estimates, this issue is considered to be outside the scope of this book.




\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/15_TORSVAG_POT.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value based on 13 years of wind data for Torsv{\aa}g Fyr station as a function of the number $n$ of data points above the threshold. The return level estimate = 49.41 at $n$ = 131, $\sigma = 5.30$}
	\label{figP15}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/16_SULA_POT.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value based on 12 years of wind data for Sula station as a function of the number $n$
	of data points above the threshold. The return level estimate = 43.42 at $n$ = 120, $\sigma = 5.49$.}
	\label{figP16}
\end{figure}
%%
\begin{figure}[!htp]
	\centerline{\includegraphics[width=2.8in]{JWEIA_Figures/17_Obrestad_POT.pdf}}
	\caption{The point estimate $\hat{\eta}^{100yr}$ of the 100-year return period value based on 12 years of wind data for Obrestad Fyr station as a function of the number $n$
	of data points above the threshold. The return level estimate = 46.1 at $n$ = 151, $\sigma = 5.47$}
	\label{figP17}
\end{figure}
%%


%%

In Tables~\ref{tab01}~-~\ref{tab04}, the 100-year return period values are listed together with the predicted 95{\%} confidence intervals for all methods and each station. In case of the annual maxima method, the 95\% confidence intervals are estimated from a parametric bootstrapping of the Gumbel estimates based on a sample of size 10,000 data sets of 13, 12 and 16 yearly extremes. For the POT method, the bootstrapped 95\% confidence intervals were estimated by using the \cite{MATLAB2009} Statistics Toolbox routine \textsf{bootstrp}. 
10,000 samples are generated by sampling with replacement from the observed exceedances above high thresholds.
%%


\begin{table}[!htp]
	\begin{center}
			\caption{Predicted 100-year return period levels for Torsv{\aa}g Fyr weather station by the ACER-method for different degrees of conditioning,
			Annual maxima and POT methods, respectively; outliers are considered true observations.}
		\begin{tabular}{l c c c}
			\toprule
			Method & Spec & \  $\hat{\eta}^{100yr}$ m/s \  & \ 95{\%} CI (${\eta}^{100yr}$) m/s \  \\ 
			\midrule
			\multirow{7}{*}{ACER, various $k$} 
			&    1   & 47.46 & (42.11, 50.71) \\ % 2.0745
			&    2   & 48.18 & (41.48, 51.31) \\ % 2.5
			&    4   & 46.96 & (42.25, 49.63) \\ % 2.5
			&    24  & 48.36 & (43.44, 51.63) \\ % 3
			&    48  & 47.54 & (43.46, 49.75) \\ % 3.5
			&    72  & 47.44 & (44.39, 48.79) \\ % 3.5
			&    96  & 48.78 & (44.53, 51.61) \\ % 3.5
			\midrule
			\multirow{2}{*}{Annual maxima} 
			& MM & 51.33 & (43.08, 61.57) \\
			%& Gumbel & 54.4 & (46.67, 71.47) \\
			& GL & 51.57 & (44.24, 60.67) \\
			\midrule
			POT & -- & 49.41 & (40.95, 59.42) \\
			\bottomrule
		\end{tabular}
		\label{tab01}
	\end{center}
\end{table}

%%

%%
\begin{table}[!htp]
	\begin{center}
			\caption{Predicted 100-year return period levels for Torsv{\aa}g Fyr weather station by the ACER-method for different degrees of conditioning, 
			Annual maxima and POT methods, respectively; outliers are rejected.}
		\begin{tabular}{l c c c}
			\toprule
			Method & Spec & \  $\hat{\eta}^{100yr}$ m/s \  & \ 95{\%} CI (${\eta}^{100yr}$) m/s \  \\ 
			\midrule
			\multirow{7}{*}{ACER, various $k$} 
			&    1     & 47.21 & (39.94, 50.60) \\ % 2.0745
			&    2     & 47.79 & (41.13, 50.93) \\ % 2.5
			&    4     & 46.32 & (42.00, 49.04) \\ % 2.5
			&    24    & 47.22 & (43.26, 50.04) \\ % 3
			&    48    & 46.38 & (43.60, 48.19) \\ % 3.5
			&    72    & 46.32 & (44.24, 47.37) \\ % 3.5
			&    96    & 47.80 & (44.45, 49.95) \\ % 3.5
			
			\midrule
			\multirow{2}{*}{Annual maxima} 
			& MM & 44.31 & (39.36, 50.39) \\
			%& Gumbel & 46.10 & (41.50, 56.53) \\
			& GL & 45.84 & (40.72, 52.41) \\
			\midrule
			POT & -- & 42.62 & (39.01, 47.31) \\
			\bottomrule
		\end{tabular}
		\label{tab02}
	\end{center}
\end{table}
%%


\begin{table}[!htp]
	\begin{center}
			\caption{Predicted 100-year return period levels for Sula weather station by the ACER-method for different degrees of conditioning, 
			Annual maxima and POT methods, respectively.}
		\begin{tabular}{l c c c}
			\toprule
			Method & Spec & \  $\hat{\eta}^{100yr}$ m/s \  & \ 95{\%} CI (${\eta}^{100yr}$) m/s \  \\ 
			\midrule
			\multirow{7}{*}{ACER, various $k$} 
			&   1     & 46.33 & (43.41, 47.77) \\ % 2.36
			&   2     & 46.81 & (44.08, 49.04) \\ % 2.5
			&   4     & 47.99 & (44.80, 50.57) \\ % 2.5
			&   24    & 46.65 & (44.10, 48.07) \\ % 3.5
			&   48    & 46.83 & (44.28, 48.03) \\ % 3.5
			&   72    & 45.80 & (43.01, 46.96) \\ % 4
			&   96    & 45.69 & (42.32, 47.01) \\ % 4
			\midrule
			\multirow{2}{*}{Annual maxima} 
			& MM & 48.66 & (41.58, 57.58) \\
			%& Gumbel & 51.05 & (44.23, 66.35) \\
			& GL & 52.90 & (44.29, 63.39) \\
			\midrule
			POT & -- & 43.42 & (39.07, 47.80) \\
			\bottomrule
		\end{tabular}
		\label{tab03}
	\end{center}
\end{table}
%%
%%

\begin{table}[!htp]
	\begin{center}
			\caption{Predicted 100-year return period levels for Obrestad Fyr weather station by the ACER-method for different degrees of conditioning, 
			Annual maxima and POT methods, respectively.}
		\begin{tabular}{l c c c}
			\toprule
			Method & Spec & \  $\hat{\eta}^{100yr}$ m/s \  & \ 95{\%} CI (${\eta}^{100yr}$) m/s \  \\ 
			\midrule
			\multirow{7}{*}{ACER, various $k$} 
			&    1     & 48.38 & (43.18, 50.74) \\ % 2.5
			&    2     & 48.11 & (42.38, 50.69) \\ %3 
			&    4     & 48.81 & (42.34, 51.59) \\ % 3
			&    24    & 47.90 & (42.87, 50.53) \\ % 3.2
			&    48    & 48.90 & (43.82, 50.72) \\ % 3.5
			&    72    & 49.47 & (44.06, 51.52) \\ % 3.7
			&    96    & 48.55 & (43.46, 49.96) \\ % 3.7
			\midrule
			\multirow{2}{*}{Annual maxima} 
			& MM & 48.59 & (42.10, 56.84) \\
			%& Gumbel & 50.63 & (44.72, 63.29) \\
			& GL & 53.79 & (46.16, 63.53) \\
			\midrule
			POT & -- & 46.10 & (41.00, 55.00) \\
			\bottomrule
		\end{tabular}
		\label{tab04}
	\end{center}
\end{table}
%%


 

\section{Extreme Value Prediction for a Narrow Band Process}\index{extreme value prediction!narrow band process}

In engineering mechanics, a classical extreme response prediction problem is the case of a lightly damped mechanical oscillator subjected to random forces. To illustrate this prediction problem, the response process of a linear mechanical oscillator driven by a Gaussian white noise will be investigated. Let $X(t)$ denote the displacement response. The dynamic model can then be expressed as, $\ddot{X}(t) + 2 \zeta \omega_e \dot{X}(t) + \omega_e^2 X(t) = W(t)$, where $\zeta$ = relative damping, $\omega_e$ = undamped eigenfrequency, and $W(t)$ = a stationary Gaussian white noise (of suitable intensity). By choosing a small value for $\zeta$, the response time series will exhibit narrow band characteristics, that is, the spectral density of the response process $X(t)$ will assume significant values only over a narrow range of frequencies. This manifests itself by producing a strong beating of the response time series, which means that the size of the response peaks will change slowly in time, see Figure~\ref{fig:ts}. A consequence of this is that neighbouring peaks are strongly correlated, and there is a conspicuous clumping of the peak values. This gives rise to the problem of accurate prediction, since the usual assumption of independent peak values is then violated.  

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=3in]{Figures_jps/Narrow_ts_peaks}%\center
\end{center}
\caption{Part of the narrow-band response time series of the linear oscillator with fully sampled and peak values indicated.}
\label{fig:ts} 
\end{figure}


Many approximations have been proposed to deal with this correlation problem, but no completely satisfactory solution has been presented. In this section it will be shown that the ACER method solves this problem efficiently and elegantly in a statistical sense. In Figure~\ref{fig:nb} are shown some of the ACER functions for the example time series. It may be verified from Figure~\ref{fig:ts} that there are approximately 32 sample points between two neighbouring peaks in the time series. To illustrate a point, the time series consisting of all sample points has been analyzed. Usually, in practice, only the time series obtained by extracting the peak values would be used for the ACER analysis of a narrow band process. In the present case, the first ACER function is then based on assuming that all the sampled data points are independent, which is obviously completely wrong. The second ACER function, which is based on counting each exceedance with an immediately preceding non-exceedance, is nothing but an upcrossing rate. Using this ACER function is largely equivalent to assuming independent peak values. It is now interesting to observe that e.g. the 25th ACER function can hardly be distinguished from the second ACER function. In fact, the ACER functions after the second do not change appreciably until one starts to approach the 32nd, which corresponds to hitting the previous peak value in the conditioning process. So, the important information concerning the dependence structure in the present time series seems to reside in the peak values, which may not be very surprising. It is seen that the ACER functions show a significant change in value as a result of accounting for the correlation effects in the time series. To verify the full dependence structure in the time series, it is necessary to continue the conditioning process down to at least the 64th ACER function. In the present case there is virtually no difference between the 32nd and the 64th, which shows that the dependence structure in this particular time series is captured almost completely by conditioning on the previous peak value. It is interesting to contrast the method of dealing with the effect of sampling frequency discussed here with that of \cite{rota00}.     


\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=3in]{Figures_jps/all_ACER_narrow}%\center
\end{center}
\caption{Comparison between ACER estimates for different degrees of
conditioning for the narrow-band time series.} \label{fig:nb}
\end{figure}

To illustrate the results obtained by extracting only the peak values from the time series, which would be the approach typically chosen in an engineering analysis, the ACER plots for this case is shown in Figure~\ref{fig:nb-peaks}. By recognizing that there is an almost one-to-one correspondence between upcrossings and peaks, it can be verified by comparing results from Figures~\ref{fig:nb} and \ref{fig:nb-peaks}, that they are in very close agreement since the second ACER function in Figure~\ref{fig:nb} corresponds to the first ACER function in Figure~\ref{fig:nb-peaks} by the observed one-to-one correspondence, and by noting that there is a factor of approximately 32 between corresponding ACER functions in the two figures. This is due to the fact that the time series of peak values contains about 32 times less data than the original time series.

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=3in]{Figures_jps/peaks_ACER_narrow}%\center
\end{center}
\caption{Comparison between ACER estimates for different degrees of
conditioning based on the time series of the peak values, cf. Figure~\ref{fig:ts}.} \label{fig:nb-peaks}
\end{figure}




