details.Rnw

%!Rnw root = Geyser-Analysis.Rnw
%!TeX root = Geyser-Analysis.Rnw

\section{Descriptive Statistics}
\label{sec:details}

\subsection{Extended summary}
\label{sec:summary}

In the following section we describe the geyser data set in more detail. In addtion to what was mentioned in Section~\ref{sec:introduction}, we want to add that we have to keep in mind that waiting time $i$ of the data set denotes the waiting time \emph{before} eruption $i$ occured. Thus waiting time $i$ incorporates duration $i-1$. We could separate the two. However, as can be seen in Table~\ref{tab:summary}, the duration is much smaller than the waiting time such that the effect on the prediction would be negligible.

\begin{table}[htbp]
\centering
\sisetup{
  table-number-alignment = right,
  table-figures-integer = 3,
  table-figures-decimal = 2
}
\begin{tabular}{r
S[table-auto-round]
S[table-auto-round]}
\toprule
& {waiting} & {duration}\\
& {[\si{\minute}]} & {[\si{\minute}]}\\
\midrule
Min & \Sexpr{min(waiting)} & \Sexpr{min(duration)}\\
1st Quantile & \Sexpr{quantile(waiting, 0.25)} & \Sexpr{quantile(duration, 0.25)}\\
Median & \Sexpr{median(waiting)} & \Sexpr{median(duration)}\\
Mean & \Sexpr{mean(waiting)} & \Sexpr{mean(duration)}\\
3rd Quantile & \Sexpr{quantile(waiting, 0.75)} & \Sexpr{quantile(duration, 0.75)}\\
Max & \Sexpr{max(waiting)} & \Sexpr{max(duration)}\\
\bottomrule
\end{tabular}
\caption{Summary of the geyser data set}
\label{tab:summary}
\end{table}

In Figure~\ref{fig:density} we see a 3 dimensional scatter plot of the next waiting time over waiting time and duration. A density estimation has been added and we see, that there are three clouds visible. These correspond exactly to the coloring in Figure~\ref{fig:cluster}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth]{density.png}
\caption{Three dimensional scatter plot with estimated density. We find three clusters corresponding to the clusters marked in Figure~\ref{fig:cluster}}
\label{fig:density}
\end{figure}