
\section{The ACER Method} \label{sec:acer}\index{ACER method}\index{extreme value distribution!ACER method} 



All calculations in this section were performed with the ACER package for Matlab \citep{ACERmanual}.


\subsection{ Application to water level measurements}

Note that for estimation of the 5, 10 and 20 year return period levels, which for all stations are in sample estimates, the ACER method will provide unbiased estimates of the exact values. For estimation of the long return period levels, the first goal is to find a tail marker $\eta_0$ that represents a sufficiently high threshold. For the ACER method, this is found where the curves of the estimated ACER functions $\hat{\varepsilon}_k(\eta)$ start behaving regularly in the sense of Eq.~\eqref{eq:ag3}. After having chosen such a point initially, one would typically go even further into the tail to verify robustness with respect to the estimated return levels. For the hourly sea level measurements studied in this chapter, it was found that a level for which the remaining data amounted to roughly 5-10\% of the total data set gave stable numerical estimates. 


\subsubsection*{Oslo}\index{water level measurements!Oslo}


The curves in the ACER plot of Figure~\ref{acer_acerplot_oslocorrected} detail the effect dependence has on the ACER function estimates for Oslo. It can be seen that the independent case of $k=1$ has a curve which stays significantly above the rest, clearly demonstrating that hourly measurements are strongly dependent. The ACER plot also shows that there is a significant diurnal dependence effect, which is to be expected. But this effect is seen to vanish at the higher levels, showing that the ACER functions coalesce in the tail for $k \ge 2$. Thus, for estimation of long return period sea levels it is advantageous to choose $k = 2$, since this case allows for the use of more data than for $k > 2$, with the potential benefit of higher accuracy as a result. 
  
With the selected estimated ACER function, i.e. $\hat{\varepsilon}_2(\eta)$, the optimized parameter estimates are $d = -4.048$, $b = 96.901$, $a = 0.037$, $c = 1.117$, which is based on a tail marker $\eta_0 = 100$. The return level estimates for Oslo are presented in Table~\ref{acer_table}.


\begin{figure}[!ht]
\centering
\subfigure[Log plot of estimated ACER functions.]{
\includegraphics[scale=0.31]{EWL/Figures/oslo_TM_100_0.pdf}
%\caption{ for Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_acerplot_oslocorrected}%
}
\subfigure[Fitted and extrapolated ACER$_2$ function.]{
\includegraphics[scale=0.31]{EWL/Figures/oslo_TM_100_1.pdf}
%\caption{ACER functions for high levels in Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_highacer_oslocorrected}%
}
\caption{ACER plots for Oslo.}
\label{acer_plots_oslocorrected}
\end{figure}
%
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\subsubsection*{Heimsjø}\index{water level measurements!Heimsjø}

Figure~\ref{acer_acerplot_heimsjo} displays well-behaved ACER functions for Heimsjø, but the plot is qualitatively quite different from the previous one for Oslo. Now, the dependence of hourly data has a smaller influence on the ACER function values. This may be due to the fact that the tidal effects are much stronger for Heimsjø than for the other two locations. Also for this location, all ACER functions for $k \ge 2$ coalesce in the tail. Therefore, $k=2$ is selected for the return level estimation. The tail marker is chosen to be $\eta_0 = 230$, which gives the following optimal values of the parameters: $d = -2.524$, $b = 154.689$, $a = 5.27 \cdot 10^{-6}$, $c = 2.755$. The return level estimates are found in Table~\ref{acer_table}.

\begin{figure}[!ht]
\centering
\subfigure[Log plot of estimated ACER functions.]{
\includegraphics[scale=0.31]{EWL/Figures/heimsjo_TM_230_0.pdf}
%\caption{ for Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_acerplot_heimsjo}%
}
\subfigure[Fitted and extrapolated ACER$_2$ function.]{
\includegraphics[scale=0.31]{EWL/Figures/heimsjo_TM_230_1.pdf}
%\caption{ACER functions for high levels in Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_highacer_heimsjo}%
}
\caption{ACER plots for Heimsjø.}
\label{acer_plots_heimsjo}
\end{figure}

%
%
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\subsubsection*{Honningsvåg}\index{water level measurements!Honningsvåg}

Figure~\ref{acer_acerplot_honning} reveals that the ACER plot for Honningsvåg is very similar to the one for Heimsjø. This may be explained by the fact that both stations have a strong tidal component. Since all ACER functions for $k \ge 2$ coalesce in the tail, $k=2$ is again selected for the return level estimation. The tail marker is chosen to be $\eta_0 = 260$, which gives the following optimal values of the parameters: $d = -3.012$, $b = 231.209$, $a = 8.15 \cdot 10^{-4}$, $c = 1.907$. The return level estimates are given in Table~\ref{acer_table}.

\begin{figure}[!ht]
\centering
\subfigure[Log plot of estimated ACER functions.]{
\includegraphics[scale=0.31]{EWL/Figures/honningsvag_TM_260_0.pdf}
%\caption{ for Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_acerplot_honning}%
}
\subfigure[Fitted and extrapolated ACER$_2$ function.]{
\includegraphics[scale=0.31]{EWL/Figures/honningsvag_TM_260_1.pdf}
%\caption{ACER functions for high levels in Oslo, with $k=1,2,3,4,5$ and 6}%
\label{acer_highacer_honning}%
}
\caption{ACER plots for Honningsvåg.}
\label{acer_plots_honning}
\end{figure}
%
%


\begin{table}[!htb]
\begin{center}
	\caption{Return level estimates and 95\% confidence intervals for the ACER model. }
	\label{acer_table}
	\vspace{1mm}
\begin{tabular}{c|c|c|c|c|c|c} 
\multicolumn{1}{c|}{}&\multicolumn{2}{c|}{Oslo}&\multicolumn{2}{c|}{Heimsjø} &\multicolumn{2}{c}{Honningsvåg}\\ 
\hline 
\multicolumn{1}{c|}{$R = 1/p$}&\multicolumn{1}{c|}{$\hat{z}_p$}&\multicolumn{1}{c|}{CI}&\multicolumn{1}{c|}{$\hat{z}_p$}&\multicolumn{1}{c|}{CI}&\multicolumn{1}{c|}{$\hat{z}_p$}&\multicolumn{1}{c}{CI} \\ 
\hline
5 & 199.6 & (191.0, 205.5) & 330.5 & (327.0, 333.6) & 351.6 & (346.4, 357.1) \\ 
10& 210.1 & (199.7, 216.9) & 336.3 & (332.4, 339.7) & 357.7 & (352.3, 363.8) \\ 
20& 220.1 & (207.7, 227.6) & 341.6 & (337.4, 345.3) & 363.3 & (357.8, 370.1)) \\ 
100& 242.3 & (224.9, 251.8) & 352.6 & (347.7, 357.0) & 375.3 & (369.6, 383.8) \\ 
200& 251.6 & (232.0, 262.0) & 357.0 & (351.8, 361.7) & 380.2 & (374.4, 389.2)\\
\hline
\end{tabular}
\end{center}
\end{table}
