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\chapter{Benchmark Problems}

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TODO: Add arnold scores

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Our goal is to develop methods that work well in
many or all cases. However, data that is based
on real observations is of particular interest.
Agroscope, the Swiss Federal Research center
for agriculture, nutrition and the environment,
has compiled some real world data. They have
kindly supplied us with the relationship matrices
so we can see how well our methods work on real data.

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The first matrix is:
\FloatBarrier
\begin{figure}[h]
    \centering
    \begin{tabular}{ c c c c c c c c c c c c }
         & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$ & $c_{11}$ \\ 
        $c_1$ & $-0.5$ & $1$ & $1$ & $1$ & $1$ & $1$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $1$ \\
        $c_2$ & $1$ & $-0.5$ & $-1$ & $1$ & $1$ & $0.5$ & $0.5$ & $-1$ & $1$ & $1$ & $1$ \\
        $c_3$ & $1$	& $-1$ & $-0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $0$ & $0$ & $0$ \\
        $c_4$ & $1$ & $1$ & $-1$ & $-0.5$ & $0$ & $0$ & $0.5$ & $-1$ & $1$ & $-1$ & $0$  \\
        $c_5$ & $1$ & $1$ & $1$ & $0$ & $-0.5$ & $0$ & $0.5$ & $1$ & $1$ & $1$ & $1$     \\
        $c_6$ & $1$	& $0.5$	& $0$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $0$ & $0$   \\
        $c_7$ & $0.5$ &	$0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ \\
        $c_8$ & $-1$ & $-1$ & $0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $0$ \\
        $c_9$ & $0.5$ & $1$ & $0$ & $1$ & $1$ & $0$ & $0.5$ & $0$ & $-0.5$ & $1$ & $0$      \\
        $c_{10}$ & $0.5$ & $1$ & $0$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $1$ & $-0.5$ & $1$\\
        $c_{11}$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$ & $0.5$ & $0$ & $0$ & $1$ & $-0.5$        \\
    \end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
    \centering
    \begin{tabular}{@{}llr@{}} \toprule
        Crop Id & Crop Name\\ \midrule
        $c_1$ & Kreuzblütler \\
        $c_2$ & Schmetterlingsblütler \\
        $c_3$ & Nachtschattengewächse \\
        $c_4$ & Kartoffel \\
        $c_5$ & Korbblütler \\
        $c_6$ & Getreide \\
        $c_7$ & Kräuter \\
        $c_8$ & Liliengewächse \\
        $c_9$ & Süssgräser \\
        $c_{10}$ & Kürbisgewächse \\
        $c_{11}$ & Gänsefussgewächse \\ \bottomrule
    \end{tabular}
\end{figure}
\FloatBarrier
and the second matrix is:
\FloatBarrier
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\begin{figure}[h]
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    \centering
    \begin{tabular}{ c c c c c c c c c c c }
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               & $c_1$	& $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$\\
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            $c_1$ & $-1$ & $0$ & $0.5$ & $-0.5$ & $1$ & $0$ & $0.5$ & $1$ & $0.5$ & $0$\\
            $c_2$ & $0$ & $-1$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $0.5$ & $0$\\
            $c_3$ & $0.5$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0.5$ & $0.5$ & $0$ & $-0.5$\\
            $c_4$ & $-0.5$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
            $c_5$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $0$\\
            $c_6$ & $0$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $1$ & $0.5$ & $0$ & $0$\\
            $c_7$ & $0.5$ & $0.5$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
            $c_8$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
            $c_9$ & $0.5$ & $0.5$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $-1$ & $0$\\
            $c_{10}$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $1$ & $0.5$ & $0.5$ & $-1$\\
    \end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
    \centering
    \begin{tabular}{@{}llr@{}} \toprule
        Crop Id & Crop Name\\ \midrule
        $c_1$ & Weisskohl \\
        $c_2$ & Karotten \\
        $c_3$ & Kartoffeln \\
        $c_4$ & Zwiebeln \\
        $c_5$ & Hanf \\
        $c_6$ & Zuckermais \\
        $c_7$ & Buschbohnen \\
        $c_8$ & Kräutermischung \\
        $c_9$ & Kopfsalat \\
        $c_{10}$ & Randen \\ \bottomrule
    \end{tabular}
\end{figure}
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\FloatBarrier
Throughout this thesis we will report for the presented
methods how well they work on these two matrices. Our
standard field size will be $15 \times 15$ pixels.