
\section{Annual Maxima Method}\index{annual maxima method}\index{extreme value distribution!annual maxima method} 


As we know from Chapter~\ref{chap:aevt}, the annual maxima (AM) method is based on the assumption that $M_N={\max}\{X_1,...,X_N\}$, where $X_1,...,X_N$  are independent observations using a block size of one year, is distributed according to the generalized extreme value (GEV) distribution. 

For the published work done on Norwegian sea levels, the shape parameter of the GEV distribution is assumed to be fixed at zero. This means that the Gumbel distribution is adopted, which can be ascertained by studying the underlying statistics and their domain of attraction. However, results for the case of a non-zero shape parameter will also be presented below. This is done because it has been argued that the GEV distribution to be used should be decided on the basis of the extreme value data \citep{cole01}. It will be shown that this can be misguided advice.  



\subsection{ Application to water level measurements}

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

Oslo has data available from all years from 1914 to 2010 -  except 1939, from which there are no available measurements. In addition, some of the years with available data have important data missing. For instance, there are only 312 data points in 1914, all from the month of December. This means that there is a very real possibility that the real annual maximum is excluded. In 1915, there is much missing data from the important autumn and winter months, where annual maxima are often found. This year is therefore also excluded from consideration. In 1972, there are no measurements in February, July, August, September and October, which means that so much data are missing that the probability that the year's maximum is left out is large. 1974 lacks data for July, August and September, and 1991 only has data for October, November and December. Both these are excluded. Finally, measurements for 2010 only go to September 16, so its maximum is also dropped from consideration. 

For the Gumbel model, the parameter estimates are $\hat{\mu}=$ 167.31 (2.074) and $\hat{\sigma}=$ 18.630 (1.497). For the GEV model, $\hat{\mu}=$ 168.21 (2.233), $\hat{\sigma}=$ 19.030 (1.579) and $\hat{\gamma} = -$ 0.089 (0.070). Standard errors are shown in parentheses.

With the data as specified above, estimates of return period levels together with maximum likelihood 95\% confidence intervals are published in Table~\ref{table_oslo} for the Gumbel and GEV models. The obtained results agree fairly well with published values collected from a report by the Norwegian Map Authority \citep{Hansen2000}, which were also based on the AM method. 


\begin{table}[!htb]
\begin{center}
	\caption{Estimates of return levels and 95\% confidence intervals for Oslo. }
	\label{table_oslo}
	\vspace{1mm}
\begin{tabular}{c|c|c|c|c} 
\multicolumn{1}{c|}{}&\multicolumn{2}{c|}{Gumbel model}&\multicolumn{2}{c}{GEV model}\\ 
 \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}\\ \hline
5 & 195.2 & (188.4, 202.1)& 194.9 & (188.7, 201.1)\\ 
10& 209.2 & (200.4, 218.0 & 207.0 & (199.1, 214.9)\\ 
20& 222.6 & (211.9, 233.4) & 217.9 & (207.4, 228.3)\\ 
100& 253.0 & (237.7, 268.2)& 240.1 & (220.5, 259.7)\\ 
200& 265.9 & (248.7, 283.2) & 248.8 & (223.9, 273.8)\\
\hline
\end{tabular}
\end{center}
\end{table}

The return level plot in Figure~\ref{gevandgumbel_rlevel_oslocorrected} shows that all points stay within the 95\% confidence intervals for both models. The two curves follow slightly different paths; the Gumbel model in Figure~\ref{gumbel_rlevel_oslocorrected} fits fairly well to all points, while the GEV model in Figure~\ref{gev_rlevel_oslocorrected} fits generally better to most points but poorly to the rightmost point. Since the estimated shape parameter is negative, the return levels of the GEV model are bounded at $\hat{\mu}-\hat{\sigma}/\hat{\gamma}=381.7$ (cm), while the Gumbel model return levels are unbounded.

\begin{figure}[!htb]\centering
\subfigure[Gumbel model]{\label{gumbel_rlevel_oslocorrected}\includegraphics[scale=0.35]{EWL/Figures/gumbel_rlevel_oslocorrected.pdf}} \hspace{0.1cm}
\subfigure[GEV model]{\label{gev_rlevel_oslocorrected}\includegraphics[scale=0.35]{EWL/Figures/gev_rlevel_oslocorrected.pdf}}\caption{Return level plots for the AM methods, Oslo.}
\label{gevandgumbel_rlevel_oslocorrected}
\end{figure}




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

Heimsjø has data for all years from 1928 to 2010. As with Oslo, some years are missing so much data that it is likely that the true annual maximum has been left out. One example is 1934, where data for all the first seven months are missing. This year is left out of the analysis. 1938 has missing data for January, February and December; important months where the real annual maximum is likely to be. 1943 lacks data for months January to April. 1959 lacks any data from September to December, and has little from February and August. Finally, 2010 lacks data for the last months of the year. All these years are therefore left out.
 
For the Gumbel model, the parameter estimates are $\hat{\mu}=$ 311.84 (1.399) and $\hat{\sigma}=$ 11.685 (0.978). For the GEV model, $\hat{\mu}=$ 312.69  (1.480), $\hat{\sigma}=$ 11.889 (1.022) and $\hat{\gamma}=-$ 0.137 (0.066). Estimates of return levels are presented in Table~\ref{table_heimsjo}.

The return level plots in Figure~\ref{gevandgumbel_rlevel_heimsjo} show the curves of the Gumbel and GEV models, together with confidence intervals and observed data points. The Gumbel model in Figure~\ref{gumbel_rlevel_heimsjo} seems to slightly overestimate the points at high levels, while the GEV model in  Figure~\ref{gev_rlevel_heimsjo} perhaps underestimates them. All points are within confidence intervals, however. The GEV model return levels are bounded at 399.7 cm.


\begin{table}[!htb]
\begin{center}
	\caption{Estimates of return levels and 95\% confidence intervals for Heimsjø. }
	\label{table_heimsjo}
	\vspace{1mm}
\begin{tabular}{c|c|c|c|c} 
\multicolumn{1}{c|}{}&\multicolumn{2}{c|}{Gumbel model}&\multicolumn{2}{c}{GEV model}\\ 
 \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}\\ \hline
5 & 329.4 & (324.8, 333.9)& 328.8 & (325.0, 332.7)\\ 
10& 338.1 & (332.3, 343.9) & 335.7 & (331.0, 340.4)\\ 
20& 346.5 & (339.5, 353.6) & 341.7 & (335.8, 347.6)\\ 
100& 365.5 & (355.5, 375.6)& 353.3 & (343.2, 363.3)\\ 
200& 373.7 & (362.4, 385.0) & 357.5 & (345.2, 369.9)\\
\hline
\end{tabular}
\end{center}
\end{table}


%
\begin{figure}[!htb]\centering
\subfigure[Gumbel model]{\includegraphics[scale=0.35]{EWL/Figures/gumbel_rlevel_heimsjo.pdf}\label{gumbel_rlevel_heimsjo}}\hspace{0.1cm}
\subfigure[GEV model]{\includegraphics[scale=0.35]{EWL/Figures/gev_rlevel_heimsjo.pdf}\label{gev_rlevel_heimsjo}}
\caption {Return level plots for the annual maxima methods, Heimsjø.}
\label{gevandgumbel_rlevel_heimsjo}
\end{figure}




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

Honningsvåg has data for all years from 1970 to 2010, except 1985. Most of the years have acceptable amounts of data, but four years are missing important data: 1970 has no data from January to May, 1988 nothing in February and very little in March, 1989 nothing in August and September and 2010 nothing from October to December. All these are excluded from the model fitting. 

For the Gumbel model, parameter estimates are $\hat{\mu}=331.08 (2.087)$ and $\hat{\sigma}=11.812 (1.458)$. For the GEV model, $\hat{\mu}=332.85 (2.285)$, $\hat{\sigma}=12.344 (1.626)$ and $\hat{\gamma}=-0.277 (0.115)$. Estimates of return levels are given in Table~\ref{table_honning}.


\begin{table}[!htb]
\begin{center}
	\caption{Estimates of return levels and 95\% confidence intervals for Honningsvåg. }
	\label{table_honning}
	\vspace{1mm}
\begin{tabular}{c|c|c|c|c} 
\multicolumn{1}{c|}{}&\multicolumn{2}{c|}{Gumbel model}&\multicolumn{2}{c}{GEV model}\\ 
 \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}\\ \hline
5 & 348.8 & (342.0, 355.6)& 348.0 & (343.2, 352.8)\\ 
10& 357.7 & (349.0, 366.3) & 353.5 & (348.3, 358.7)\\ 
20& 366.2 & (355.6, 376.7) & 357.8 & (351.7, 363.9)\\ 
100& 385.4 & (370.4, 400.4)& 365.0 & (355.3, 374.7)\\ 
200& 393.6 & (376.7, 410.6) & 367.2 & (355.6, 378.7)\\
\hline
\end{tabular}
\end{center}
\end{table}


%
The probability and quantile plots show that there are fewer points available for the model estimation than for Oslo and Heimsjø. The fit is still quite good overall, and similar to the other two locations - even though only a few points are on the line, the points seem to follow a sort of oscillating S shape around the optimal straight line. 

The return level plots in Figure~\ref{gevandgumbel_rlevel_honning} show the trend of the Gumbel and GEV models. Again it can be seen that the fit is not impressive, although the data points stay within the confidence bounds. As for Oslo and Heimsjø, the estimated shape parameter is negative, $\hat{\gamma}=-0.2771$, meaning that the Gumbel model gives higher estimates for the return levels for high return periods. Thus, it would appear that the higher return levels might be overestimated by the Gumbel model in Figure~\ref{gumbel_rlevel_honning} and somewhat underestimated by the GEV in Figure~\ref{gev_rlevel_honning}. The GEV return levels are bounded at 377.4 cm, significantly below even the 100 year return level of the Gumbel model. This is clearly not a sensible situation, and points to the hazard of using essentially curve fitting to decide on which GEV to use for estimating long return period sea levels. 
%
\begin{figure}[!htb]
\centering
\subfigure[Gumbel model]{
\includegraphics[scale=0.35]{EWL/Figures/gumbel_rlevel_honning.pdf}
\label{gumbel_rlevel_honning}
}
\subfigure[GEV model]{
\includegraphics[scale=0.35]{EWL/Figures/gev_rlevel_honning.pdf}
\label{gev_rlevel_honning}
}
\caption {Return level plots for the annual maxima methods, Honningsvåg.}
\label{gevandgumbel_rlevel_honning}
\end{figure}
%