\section{Statistical uncertainty}
\label{sec:stat}

\subsection{Nominal analysis}

This section will detail the estimation of the statistical uncertainty. The estimation relies on bootstrapping to generate many pseudo data set with varied statistics.
Figure~\ref{fig:stat} shows a diagram of the procedure.
Each pseudo data set is generated from varying the statistics of the original positron counts histogram of all the 24 calorimeters combined.
The number of entries $N_i$ in each bin $i$ is varied randomly by either + or - $\sqrt{N_i}$, i.e., the variation follows a Poisson statistics (the positron counts
histogram corresponds to a counting experiment).
The fast rotation analysis is performed on each of the many pseudo data in order to get an ensemble of results from which is estimated the statistical uncertainty.
The definition of the background, i.e. which data point are defined as background, is fixed for analyzing all the pseudo data set.
This is necessary to avoid a systematic effect due to the background definition\footnote{This effect would translate into multiple peaks in the distribution of the results. These
peaks would have the same standard deviation but different average values.}.
The parameters \tz, \ts\ and \tm\ are optimized for each pseudo data set.
Figure~\ref{fig:stat_nominal} shows the ensemble of results for the nominal fast rotation analysis.
The statistical uncertainty is taken as the standard deviation of the distribution of results.
Table~\ref{tab:stat} summarizes the numbers. The statistical uncertainty appears to be very small and
will be negligible in comparison to the systematic uncertainties (see Sec.~\ref{sec:syst}).
Figure~\ref{fig:stat_corr} shows the statistical correlation distributions between the results. It appears that $x_e$ is very strongly
anti-correlated with \tz. This is explained by the fact that changing \tz\ skews the cosine Fourier transform to the right or to the left,
therefore shifting its average value one way or the other.

\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline\hline
& $t_0$ [ns] & $x_{e}$ [mm] & $\sigma$ [mm] & $C_{E}$ [ppb]\\ \hline
nominal         &  0.011 &  0.008 &  0.007 &  0.72 \\
per-bunch       &  0.030 &  0.022 &  0.018 &  1.86 \\
per-calorimeter &  0.055 &  0.041 &  0.033 &  3.43 \\
\hline
\hline
\end{tabular}
\caption{Statistical uncertainty on the results of the nominal, per-bunch and per-calorimeter fast rotation analysis.}
\label{tab:stat}
\end{center}
\end{table}

\begin{figure}[bt]
\centering
\includegraphics[width=0.7\textwidth]{fig/stat}
\caption{Diagram of the procedure for estimating the statistical uncertainty on the fast rotation results.}
\label{fig:stat}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_t0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_xe}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_width}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_ce}}
\caption{Analysis results of the pseudo data for the nominal analysis: (a) optimized \tz, \mbox{(b) $x_e$}, (c) $\sigma$, and (d) $C_E$.
The title of each histogram corresponds to the mean and standard deviation of the distribution.}
\label{fig:stat_nominal}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_ceVSwithVSxe}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_t0VSwithVSxe}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_widthVSt0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_xeVSt0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_widthVSxe}}
\caption{Statistical correlation between: (a) $C_E$, $\sigma$ and $x_e$, (b) $\Delta t_0 = <t_0>-t_0$, $\sigma$ and $x_e$, 
(c) $\sigma$ and \tz, (d) $x_e$ and \tz, and (e) $\sigma$ and $x_e$,
The title of the 2D histograms corresponds to the correlation coefficient between the two variables.}
\label{fig:stat_corr}
\end{figure}

\subsection{Per-bunch analysis}
\label{sec:stat_perbunch}

The procedure above is used to estimate the statistical uncertainty of the results per-bunch with the 24 calorimeter combined, i.e., the statistics
is an $8^\text{th}$ of the nominal statistics..
The estimation is performed using bunch \#0. It is assumed that the statistical uncertainty of the other 7 bunches is the same.
Figure~\ref{fig:stat_perbunch} shows the distributions of the results. The statistical uncertainty scales as expected: $\sqrt{8}$ times bigger 
than the values for the nominal analysis which accounts for analyzing 1 bunch versus analyzing 8 combined.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perBunch_t0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perBunch_xe}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perBunch_width}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perBunch_ce}}
\caption{Analysis results of the pseudo data for bunch \#0: (a) optimized \tz, (b) $x_e$, (c) $\sigma$, and (d) $C_E$.
The title of each histogram corresponds to the mean and standard deviation of the distribution.}
\label{fig:stat_perbunch}
\end{figure}

\subsection{Per-calorimeter analysis}
\label{sec:stat_percalo}

The procedure above is used to estimate the statistical uncertainty of the results per-calorimeter with the 8 bunches combined, i.e., the statistics
is a $24^\text{th}$ of the nominal statistics.
The estimation is performed using calorimeter \#1. It is assumed that the statistical uncertainty of the other 23 calorimeters is the same.
Figure~\ref{fig:stat_percalo} shows the distributions of the results. The statistical uncertainty scales as expected: $\sqrt{24}$ times bigger 
than the values for the nominal analysis which accounts for analyzing 1 calorimeter versus analyzing 24 combined.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perCalo_t0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perCalo_xe}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perCalo_width}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/stat_perCalo_ce}}
\caption{Analysis results of the pseudo data for calorimeter \#1: (a) optimized \tz, \mbox{(b) $x_e$}, \mbox{(c) $\sigma$} and (d) $C_E$.
The title of each histogram corresponds to the mean and standard deviation of the distribution.}
\label{fig:stat_percalo}
\end{figure}

\clearpage
