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version 1.2, 2011/05/18 21:54:06
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version 1.3, 2011/06/01 01:20:54
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| \newpage |
\newpage |
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| \section{\label{sub:NN}Neural Network Analysis} |
\section{\label{sub:NN}Neural Network Analysis} |
| |
|
| \subsection{\label{sub:Variables}Variables for NN training} |
\subsection{\label{sub:Variables}Variables for NN training} |
| |
|
| \noindent Following the same procedure as in the previous analysis, we determine the content |
\noindent Following the same procedure as in the p17 analysis, we determine the content |
| of signal and background in the preselected sample, increase signal/background rate |
of signal and background in the preselected sample, increase signal/background rate |
| and from this, measure the cross-section. |
and from this, measure the cross section. |
| The procedure adopted in the p17 analysis was feed a set of topological variables into an |
In p17, an artificil neural network based on topological characteristics of an event was used to |
| artificial neural network in order to provide the best possible separation between |
extract signal from a background-enriched region. As before, the criteria used in choosing the variables were: |
| signal and background. As before, the criteria for choosing such variables were: power of discrimination |
power of discrimination and $\tau$-uncorrelated variables. The following variables were considered: |
| and $\tau$-uncorrelated variables. The set is presented below: |
|
| |
|
| \begin{itemize} |
\begin{itemize} |
| \item \textit{\textbf{$H_{T}$}} - the scalar sum of all jet's $p_{T}$ (here and below including $\tau$ lepton candidates). |
\item \textit{\textbf{$H_{T}$}} - The scalar sum of all jet $p_{T}$'s (here and below including $\tau$ lepton candidates). |
| |
For $H_{T}$ values above $\sim$ 200 GeV we observed a dominance of signal over background. |
| |
|
| \item \textit{\textbf{$\not\!\! E_{T}$ significance}} - As being the variable that provides the best signal-background |
\item \textit{\textbf{$\not\!\! E_{T}$ significance}} - It is computed from calculated resolutions of |
| separation we decided to optimize it. |
physical objects (jets, electrons, muons and unclustered energy) \cite{p17_note,METsig}. |
| |
It was chosen to be used and optimized due to its good signal-background discrimination power. |
| |
|
| \item \textit{\textbf{Aplanarity}} \cite{p17topo} - the normalized momentum tensor is defined as |
\item \textit{\textbf{Aplanarity}} \cite{p17topo} - the normalized momentum tensor is defined as |
| |
|
| \begin{center} |
\begin{center} |
| \begin{equation} |
\begin{equation} |
| {\cal M} = \frac{\sum_{o}p^{o}_{i}p^{o}_{j}}{\sum_{o}|\overrightarrow{p^{o}}|} |
{\cal M}_{ab} \equiv \frac{\sum_{i}p_{ia}p_{ib}}{\sum_{i}p^{2}_{i}} |
| \label{tensor} |
\label{tensor} |
| \end{equation} |
\end{equation} |
| \end{center} |
\end{center} |
| |
|
| \noindent where $\overrightarrow{p^{0}}$ is the momentum-vector of a reconstructed object $o$ |
\noindent where $p_{i}$ is the momentum-vector |
| and $i$ and $j$ are cartesian coordinates. From the diagonalization of $\cal M$ we find three eigenvalues |
and teh index $i$ runs over all the jets and the $W$. From the diagonalization of $\cal M$ we find three eigenvalues |
| $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ with the constraint $\lambda_{1} + \lambda_{2} + \lambda_{3} = 1$. |
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ with the constraint $\lambda_{1} + \lambda_{2} + \lambda_{3} = 1$. |
| The aplanarity {\cal A} is given by {$\cal A$} = $\frac{3}{2}\lambda_{3}$ and measures the flatness of an event. |
The aplanarity is defined as {$\cal A$} = $\frac{3}{2}\lambda_{3}$ and measures the flatness of an event. |
| Hence, it is defined in the range $0 \leq {\cal M} \leq 0.5$. Large values of {$\cal A$} correspond to more spherical events, |
It assumes values in the range $0 \leq {\cal A} \leq 0.5$. |
| like $t\bar{t}$ events for instance, since they are typical of decays of heavy objects. On the other hand, |
It was chosen to be used in the NN due to the fact that large values of {$\cal A$} correspond to more spherical events, |
| both QCD and $W + \mbox{jets}$ events are more planar since jets in these events are primarily due to |
like $t\bar{t}$ events for instance, since they are typical of cascade decays of heavy objects. On the other hand, |
| |
both QCD and $W + \mbox{jets}$ events tend to be more collinear since jets in these events are primarily due to |
| initial state radiation. |
initial state radiation. |
| |
|
| \item \textit{\textbf{Sphericity}} \cite{p17topo} - being defined as {$\cal S$} = $\frac{3}{2}(\lambda_{2} + \lambda_{3})$, |
\item \textit{\textbf{Sphericity}} \cite{p17topo} - Defined as {$\cal S$} = $\frac{3}{2}(\lambda_{2} + \lambda_{3})$, |
| and having a range $0 \leq {\cal S} \leq 1.0$, sphericity is a measure of the summed $p^{2}_{\perp}$ with |
and ranges as $0 \leq {\cal S} \leq 1.0$, sphericity is a measure of the summed $p^{2}_{\perp}$ |
| respect to the event axis. In this sense a 2-jets event corresponds to {$\cal S$} $\approx 0$ and an isotropic event |
More isotropic events have {$\cal S$} $\approx 1$ while less isotropic ones have {$\cal S$} $\approx 0$. |
| {$\cal S$} $\approx 1$. $t\bar{t}$ events are very isotropic as they are typical of the decays of heavy objects |
Sphericity is a good discrminator since $t\bar{t}$ events are very isotropic as they are typical of the decays of heavy objects |
| and both QCD and $W + \mbox{jets}$ events are less isotropic due to the fact that jets in these events come |
and both QCD and $W + \mbox{jets}$ events are less isotropic due to the fact that jets in these events come |
| primarily from initial state radiation. |
primarily from initial state radiation. |
| |
|
|
Line 48 $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma
|
Line 50 $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma
|
| where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (172.4 |
where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (172.4 |
| GeV and 81.02 GeV respectively) and resolution values (19.4 GeV and 8.28 |
GeV and 81.02 GeV respectively) and resolution values (19.4 GeV and 8.28 |
| GeV respectively). $M_{3j}$ and $M_{2j}$ are invariant masses composed |
GeV respectively). $M_{3j}$ and $M_{2j}$ are invariant masses composed |
| of the jet combinations. We choose combination that minimizes $L$. |
of 2- and 3-jet combinations. We choose combination that minimizes $L$. |
| |
|
| \item \textit{\textbf{Centrality}}, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum |
\item \textit{\textbf{Centrality}}, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum |
| of energies of the jets. |
of energies of the jets. Used as discrimination variable since highe values ($\sim$ 1.0) are |
| |
more signal-dominated while low values ($\sim$ 0) are more background-dominated. |
| |
|
| \item \textit{\textbf{$\cos(\theta*)$}} - The angle between the beam axis and the |
\item \textit{\textbf{$\cos(\theta*)$}} - The angle between the beam axis and the |
| highest-$p_T$ jet in the rest frame of all the jets in the event. |
highest-$p_T$ jet in the rest frame of all the jets in the event. $t\bar{t}$ events tend to have |
| \item \textit{\textbf{$\sqrt(s)$}} - The invariant mass of all jets and $\tau$s in the event. |
a lower ($\sim$ 0) $\cos(\theta*)$ values. This motivated its choice. |
| |
|
| |
\item \textit{\textbf{$M_{jj\tau}$}} - The invariant mass of all jets and $\tau$s in the event. |
| |
|
| \end{itemize} |
\end{itemize} |
| |
|
|
Line 62 The chosen variables are in the end a co
|
Line 68 The chosen variables are in the end a co
|
| analysis: use events from the QCD-enriched |
analysis: use events from the QCD-enriched |
| loose-tight sample to model QCD events in the signal-rich sample, and use |
loose-tight sample to model QCD events in the signal-rich sample, and use |
| a b-tag veto sample as an independent control sample to check the validity of such |
a b-tag veto sample as an independent control sample to check the validity of such |
| background modeling. |
background modeling. Plots of all variables described above are found in Appendix \ref{app:discri_var}. |
| |
|
| %\clearpage |
%\clearpage |
| |
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|
Line 70 background modeling.
|
Line 76 background modeling.
|
| For training the Neural Network we used the Multilayer Perceptron algorithm, as described in |
For training the Neural Network we used the Multilayer Perceptron algorithm, as described in |
| \cite{MLPfit}. As explained before in Section \ref{sub:Results-of-the}, the first 1400000 |
\cite{MLPfit}. As explained before in Section \ref{sub:Results-of-the}, the first 1400000 |
| events in the ``loose-tight'' sample were used as background |
events in the ``loose-tight'' sample were used as background |
| for NN training for taus types 1 and 2, and the first 600000 of the same sample for NN training for type 3 taus. |
for NN training for taus of Types 1 and 2, and the first 600000 of the same sample for NN training for type 3 taus. |
| This means that different tau types are being treated separately in the topological NN. |
|
| In both cases 1/3 of the Alpgen sample of $t\bar{t} \rightarrow \tau +jets$ was used for NN training and 2/3 of it |
In both cases 1/3 of the Alpgen sample of $t\bar{t} \rightarrow \tau +jets$ was used for NN training and 2/3 of it |
| for the measurement. |
for the measurement. |
| When doing the measurement later on (Section \ref{sub:xsect}) we pick the tau with the highest $NN(\tau)$ |
When doing the measurement later on (Section \ref{sub:xsect}) we pick the tau with the highest $NN(\tau)$ |
|
Line 88 the signal-enriched region.
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Line 93 the signal-enriched region.
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|
| |
|
| \begin{figure}[h] |
\begin{figure}[h] |
| \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type2_tauQCD.eps} |
\includegraphics[scale=0.49]{plots/SetI_NNout_SM_type2_tauQCD.eps} |
| \caption{Training of topological Neural Network output for type 1 and 2 $\tau$ channel. |
\caption{Training of topological Neural Network output for Type 1 and 2 $\tau$ channel combined. |
| Upper left: relative impact of each of the input variables; upper right: topological structure; |
Upper left: relative impact of each of the input variables; upper right: relative weights |
| lower right: final signal-background separation of the method; lower left: convergence curves.} |
of the synaptic connections of the trained network; |
| |
lower left: convergence curves; lower right: the output distribution of signal and background |
| |
test samples after training.} |
| \label{fig:nnout_type2_training} |
\label{fig:nnout_type2_training} |
| \end{figure} |
\end{figure} |
| |
|
| \newpage |
%\newpage |
| |
|
| |
|
| \begin{figure}[h] |
\begin{figure}[h] |
| \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type3_tauQCD.eps} |
\includegraphics[scale=0.49]{plots/SetI_NNout_SM_type3_tauQCD.eps} |
| \caption{Training of topological Neural Network output for type 3 $\tau$ channel. |
\caption{Training of topological Neural Network output for type 3 $\tau$ channel. |
| Upper left: relative impact of each of the input variables; upper right: topological structure; |
Upper left: relative impact of each of the input variables; upper right: relative weights |
| lower right: final signal-background separation of the method; lower left: convergence curves.} |
of the synaptic connections of the trained network; |
| |
lower left: convergence curves; lower right: the output distribution of signal and background |
| |
test samples after training.} |
| \label{fig:nnout_type3_training} |
\label{fig:nnout_type3_training} |
| \end{figure} |
\end{figure} |
| |
|
|
Line 148 level in order to have a consistent cut
|
Line 157 level in order to have a consistent cut
|
| Below we describe how we split this part of the analysis into two parts: |
Below we describe how we split this part of the analysis into two parts: |
| |
|
| \begin{enumerate} |
\begin{enumerate} |
| \item {\bf Set optimization:} We applied an arbitrary cut on $\not\!\! E_{T}$ significance of $\geq$ 4.0 and |
\item {\bf Set optimization:} We applied an ``reasonable'' cut on $\not\!\! E_{T}$ significance of $\geq$ 4.0 and |
| varied the set of varibles going into NN training |
varied the set of varibles going into NN training. |
| \item {\bf $\not\!\! E_{T}$ significance optimization:} After chosing the best set based on the lowest RMS, |
\item {\bf $\not\!\! E_{T}$ significance optimization:} After chosing the best set based on the lowest RMS of the |
| we then varied the $\not\!\! E_{T}$ significance cut |
figure of merith used (see Eq. \ref{merit}), we then optimized the $\not\!\! E_{T}$ significance cut. |
| \end{enumerate} |
\end{enumerate} |
| |
|
| For this part of the analysis we present the sets of variables that were taken into account to perform the NN traning |
For this part of the analysis we present the sets of variables that were taken into account to perform the NN traning: |
| \begin{itemize} |
\begin{itemize} |
| \item \textit{\textbf{Set I}} : {$H_{T}$}, aplan (aplanarity), sqrts ($\sqrt{s}$) |
\item \textit{\textbf{Set 1}} : {$H_{T}$}, aplan (aplanarity), Mjjtau ($M_{jj\tau}$) |
| \item \textit{\textbf{Set II}} : {$H_{T}$}, aplan, cent (centrality) |
\item \textit{\textbf{Set 2}} : {$H_{T}$}, aplan, cent (centrality) |
| \item \textit{\textbf{Set III}} : {$H_{T}$}, aplan, spher (spherecity) |
\item \textit{\textbf{Set 3}} : {$H_{T}$}, aplan, spher (spherecity) |
| \item \textit{\textbf{Set IV}} : {$H_{T}$}, cent, spher |
\item \textit{\textbf{Set 4}} : {$H_{T}$}, cent, spher |
| \item \textit{\textbf{Set V}} : aplan, cent, spher |
\item \textit{\textbf{Set 5}} : aplan, cent, spher |
| \item \textit{\textbf{Set VI}} : {$H_{T}$}, aplan, sqrts, spher |
\item \textit{\textbf{Set 6}} : {$H_{T}$}, aplan, Mjjtau, spher |
| \item \textit{\textbf{Set VII}} : {$H_{T}$}, aplan, sqrts, cent |
\item \textit{\textbf{Set 7}} : {$H_{T}$}, aplan, Mjjtau, cent |
| \item \textit{\textbf{Set VIII}} : {$H_{T}$}, aplan, sqrts, costhetastar ($cos(\theta^{*})$ |
\item \textit{\textbf{Set 8}} : {$H_{T}$}, aplan, Mjjtau, costhetastar ($cos(\theta^{*})$) |
| \item \textit{\textbf{Set IX}} : {$H_{T}$}, aplan, sqrts, cent, spher |
\item \textit{\textbf{Set 9}} : {$H_{T}$}, aplan, Mjjtau, cent, spher |
| \item \textit{\textbf{Set X}} : {$H_{T}$}, aplan, sqrts, cent, costhetastar |
\item \textit{\textbf{Set 10}} : {$H_{T}$}, aplan, Mjjtau, cent, costhetastar |
| \item \textit{\textbf{Set XI}} : {$H_{T}$}, aplan, sqrts, spher, costhetastar |
\item \textit{\textbf{Set 11}} : {$H_{T}$}, aplan, Mjjtau, spher, costhetastar |
| \item \textit{\textbf{Set XII}} : metl, {$H_{T}$}, aplan, sqrts |
\item \textit{\textbf{Set 12}} : METsig ($\not\!\! E_{T}$ significance), {$H_{T}$}, aplan, Mjjtau |
| \item \textit{\textbf{Set XIII}} : metl, {$H_{T}$}, aplan, cent |
\item \textit{\textbf{Set 13}} : METsig, {$H_{T}$}, aplan, cent |
| \item \textit{\textbf{Set XIV}} : metl, {$H_{T}$}, aplan, spher |
\item \textit{\textbf{Set 14}} : METsig, {$H_{T}$}, aplan, spher |
| \item \textit{\textbf{Set XV}} : metl, {$H_{T}$}, cent, spher |
\item \textit{\textbf{Set 15}} : METsig, {$H_{T}$}, cent, spher |
| \item \textit{\textbf{Set XVI}} : metl, {$H_{T}$}, aplan |
\item \textit{\textbf{Set 16}} : METsig, {$H_{T}$}, aplan |
| \item \textit{\textbf{Set XVII}} : metl, {$H_{T}$}, sqrts |
\item \textit{\textbf{Set 17}} : METsig, {$H_{T}$}, Mjjtau |
| \item \textit{\textbf{Set XVIII}} : metl, aplan, sqrts |
\item \textit{\textbf{Set 18}} : METsig, aplan, Mjjtau |
| \item \textit{\textbf{Set XIX}} : metl, {$H_{T}$}, cent |
\item \textit{\textbf{Set 19}} : METsig, {$H_{T}$}, cent |
| \item \textit{\textbf{Set XX}} : metl, {$H_{T}$}, aplan, sqrts, cent |
\item \textit{\textbf{Set 20}} : METsig, {$H_{T}$}, aplan, Mjjtau, cent |
| \item \textit{\textbf{Set XXI}} : metl, {$H_{T}$}, aplan, cent, spher |
\item \textit{\textbf{Set 21}} : METsig, {$H_{T}$}, aplan, cent, spher |
| \item \textit{\textbf{Set XXII}} : metl, {$H_{T}$}, aplan, sqrts, spher |
\item \textit{\textbf{Set 22}} : METsig, {$H_{T}$}, aplan, Mjjtau, spher |
| \item \textit{\textbf{Set XXIII}} : metl, {$H_{T}$}, aplan, sqrts, costhetastar |
\item \textit{\textbf{Set 23}} : METsig, {$H_{T}$}, aplan, Mjjtau, costhetastar |
| \item \textit{\textbf{Set XXIV}} : metl, sqrts, cent, spher, costhetastar |
\item \textit{\textbf{Set 24}} : METsig, Mjjtau, cent, spher, costhetastar |
| \item \textit{\textbf{Set XXV}} : metl, {$H_{T}$}, cent, spher, costhetastar |
\item \textit{\textbf{Set 25}} : METsig, {$H_{T}$}, cent, spher, costhetastar |
| \item \textit{\textbf{Set XXVI}} : metl, aplan, cent, spher, costhetastar |
\item \textit{\textbf{Set 26}} : METsig, aplan, cent, spher, costhetastar |
| \item \textit{\textbf{Set XXVII}} : metl, {$H_{T}$}, aplan, cent, costhetastar |
\item \textit{\textbf{Set 27}} : METsig, {$H_{T}$}, aplan, cent, costhetastar |
| \item \textit{\textbf{Set XXVIII}} : {$H_{T}$}, aplan, topmassl |
\item \textit{\textbf{Set 28}} : {$H_{T}$}, aplan, topmassl |
| \item \textit{\textbf{Set XXIX}} : {$H_{T}$}, aplan, sqrts, topmassl |
\item \textit{\textbf{Set 29}} : {$H_{T}$}, aplan, Mjjtau, topmassl |
| \item \textit{\textbf{Set XXX}} : {$H_{T}$}, aplan, sqrts, cent, topmassl |
\item \textit{\textbf{Set 30}} : {$H_{T}$}, aplan, Mjjtau, cent, topmassl |
| \item \textit{\textbf{Set XXXI}} : {$H_{T}$}, aplan, sqrts, costhetastar, topmassl |
\item \textit{\textbf{Set 31}} : {$H_{T}$}, aplan, Mjjtau, costhetastar, topmassl |
| \item \textit{\textbf{Set XXXII}} : metl, {$H_{T}$}, topmassl, aplan, sqrts |
\item \textit{\textbf{Set 32}} : METsig, {$H_{T}$}, topmassl, aplan, Mjjtau |
| \item \textit{\textbf{Set XXXIII}} : metl, spher, costhetastar, aplan, cent |
\item \textit{\textbf{Set 33}} : METsig, spher, costhetastar, aplan, cent |
| % \item \textit{\textbf{Set XXXIV}} : metl, spher, sqrts, topmassl, ktminp |
% \item \textit{\textbf{Set XXXIV}} : metl, spher, Mjjtau, topmassl, ktminp |
| \end{itemize} |
\end{itemize} |
| |
|
| |
P17 tried only three different sets among hundreds of possible combinations. We believe that the |
| |
33 sets tested above suffice in giving an optimal result. |
| The criteria used for making a decision on which variable should be used follow: |
The criteria used for making a decision on which variable should be used follow: |
| \begin{itemize} |
\begin{itemize} |
| \item No more than 5 variables to keep NN simple and stable. More variables leads to instabilities (different |
\item No more than 5 variables to keep NN simple and stable. More require larger training samples. |
| result after each retraining) and require larger training samples. |
\item We want to use METsig variable, since it's the one providing best discrimination. |
| % \item We want to use $metl$ (\met significance) variable, since it's the one providing best discrimination. |
\item We do not want to use highly correlated variables in same NN. Such as $H_{T}$ and jet $p_{T}$'. |
| \item We do not want to use highly correlated variables in same NN. |
|
| % \item We can not use tau-based variables. |
% \item We can not use tau-based variables. |
| \item We want to use variables with high discriminating power. |
\item We want to use variables with high discriminating power. |
| \end{itemize} |
\end{itemize} |
| |
|
| In order to make the decision about which of these 11 choices is the optimal we created an ensemble of |
In order to make the decision about which of these 33 choices is the optimal we created an ensemble of |
| 20000 pseudo-datasets each containing events randomly (according to a Poisson distribution) picked |
20000 pseudo-datasets each containing events randomly (according to a Poisson distribution) picked |
| from QCD, EW and $\ttbar$ templates. Each of these datasets was treated like real data, meaning applying all |
from QCD, EW and $\ttbar$ templates. Each of these datasets was treated like real data, meaning applying all |
| the cuts and doing the shape fit of event topological NN. QCD templates for fit were made from the same |
the cuts and doing the shape fit of event topological NN. QCD templates for fit were made from the same |
| ``loose-tight $\tau$ sample'' from which the QCD component of the ``data'' was drawn. |
``loose-tight $\tau$ sample'' from which the QCD component of the ``data'' was drawn. |
| The figure of merit chosen is given by Equation \ref{merit} below: |
We used the folloing quantity as the figure of merit: |
| |
|
| \begin{equation} |
\begin{equation} |
| f = \displaystyle \frac{(N_{fit} - N_{true})}{N_{true}} |
f = \displaystyle \frac{(N_{fit} - N_{true})}{N_{true}} |
|
Line 218 f = \displaystyle \frac{(N_{fit} - N_{tr
|
Line 228 f = \displaystyle \frac{(N_{fit} - N_{tr
|
| \noindent where $N_{fit}$ is the number of $t\bar{t}$ pairs given by |
\noindent where $N_{fit}$ is the number of $t\bar{t}$ pairs given by |
| the fit and $N_{true}$ is the number of $t\bar{t}$ pairs from the Poisson distribution. |
the fit and $N_{true}$ is the number of $t\bar{t}$ pairs from the Poisson distribution. |
| In both Set and $\not\!\! E_{T}$ significance optimization, the lowest RMS was used |
In both Set and $\not\!\! E_{T}$ significance optimization, the lowest RMS was used |
| to caracterize which configuration is the best in each case. |
to characterize which configuration is the best in each case. |
| |
|
| |
|
| The plots showing results concerning the set optimizations are found in Appendix \ref{app:set_opt} and are summarized |
The plots showing results concerning the set optimization are found in Appendix \ref{app:set_opt} and are summarized |
| in Table \ref{setopt_table} below, where each RMS and mean are shown. |
in Table \ref{setopt_table} below, where each RMS and mean are shown. The parenthesis after each set ID show the number of |
| For NN training is standard to choose the number of hidden nodes as being twice |
|
| the number the number of variables used for the training. The parenthesis after each set ID show the number of |
|
| hidden nodes in NN training. |
hidden nodes in NN training. |
| |
|
| \begin{table}[htbp] |
\begin{table}[htbp] |
|
Line 307 The number in parenthesis refers to numb
|
Line 315 The number in parenthesis refers to numb
|
| \label{setopt_table} |
\label{setopt_table} |
| \end{table} |
\end{table} |
| |
|
| From Table \ref{setopt_table} we see that Set I has the lowest RMS, thus we chose it |
From Table \ref{setopt_table} we see that Set 32 has the lowest RMS, thus we chose it |
| as the set to be used in $\not\!\! E_{T}$ significance optimization part, whose results are |
as the set to be used in $\not\!\! E_{T}$ significance optimization part, whose results are |
| shown in Appendix \ref{app:metl_opt} and then summarized in Table \ref{metlopt_table} below |
shown in Appendix \ref{app:metl_opt} and then summarized in Table \ref{metlopt_table} below |
| |
|
| \begin{table}[htbp] |
\begin{table}[htbp] |
| \begin{tabular}{|c|r|r|r|} \hline |
\begin{tabular}{|c|r|r|r|r|} \hline |
| Set of variables & $\not\!\! E_{T}$ significance cut & RMS & \multicolumn{1}{c|}{mean} \\ \hline |
Set 32 & Number of hidden nodes &$\not\!\! E_{T}$ significance cut & RMS & \multicolumn{1}{c|}{mean} \\ \hline |
| |
|
| \hline |
\hline |
| |
|
|
Line 326 Set of variables & $\not\!\! E_{T}$ sig
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Line 334 Set of variables & $\not\!\! E_{T}$ sig
|
| |
|
| %Set6(10) & \multicolumn{1}{c|}{2.5} & \multicolumn{1}{c|}{0.2021} \\ \hline |
%Set6(10) & \multicolumn{1}{c|}{2.5} & \multicolumn{1}{c|}{0.2021} \\ \hline |
| |
|
| Set32(10) & \multicolumn{1}{c|}{3.0} & \multicolumn{1}{c|}{0.1507} & \multicolumn{1}{c|}{0.0157}\\ \hline |
1 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{3.0} & \multicolumn{1}{c|}{0.1507} & \multicolumn{1}{c|}{0.0157}\\ \hline |
| |
|
| Set32(10) & \multicolumn{1}{c|}{3.5} & \multicolumn{1}{c|}{0.1559} & \multicolumn{1}{c|}{0.0189}\\ \hline |
2 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{3.5} & \multicolumn{1}{c|}{0.1559} & \multicolumn{1}{c|}{0.0189}\\ \hline |
| |
|
| Set32(10) & \multicolumn{1}{c|}{4.0} & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline |
3 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{4.0} & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline |
| |
|
| Set32(10) & \multicolumn{1}{c|}{4.5} & \multicolumn{1}{c|}{0.1511} & \multicolumn{1}{c|}{0.0153}\\ \hline |
4 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{4.5} & \multicolumn{1}{c|}{0.1511} & \multicolumn{1}{c|}{0.0153}\\ \hline |
| |
|
| Set32(10) & \multicolumn{1}{c|}{5.0} & \multicolumn{1}{c|}{0.1552} & \multicolumn{1}{c|}{0.0205}\\ \hline |
5 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{5.0} & \multicolumn{1}{c|}{0.1552} & \multicolumn{1}{c|}{0.0205}\\ \hline |
| |
|
| %Set6(10) & \multicolumn{1}{c|}{5.5} & \multicolumn{1}{c|}{0.4008} \\ \hline |
%Set6(10) & \multicolumn{1}{c|}{5.5} & \multicolumn{1}{c|}{0.4008} \\ \hline |
| \end{tabular} |
\end{tabular} |
|
Line 346 The number in parenthesis refers to numb
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Line 354 The number in parenthesis refers to numb
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| |
|
| Combined results from Tables \ref{setopt_table} and \ref{metlopt_table} show that the best configuration found |
Combined results from Tables \ref{setopt_table} and \ref{metlopt_table} show that the best configuration found |
| was Set I with $\not\!\! E_{T}$ significance $\geq$ 4.0. Therefore, this was the |
was Set 32 with $\not\!\! E_{T}$ significance $\geq$ 4.0. Therefore, this was the |
| configuration used to perform the cross-section measurement.Figure \ref{fig:METsig_RMS} shows the variation of the RMS as function |
configuration used to perform the cross section measurement.Figure \ref{fig:METsig_RMS} shows the variation of the RMS as function |
| of the $\not\!\! E_{T}$ significance we applied. |
of the $\not\!\! E_{T}$ significance we applied. |
| |
|
| \begin{figure}[b] |
\begin{figure}[h] |
| \includegraphics[scale=0.4]{plots/METsig-RMS.eps} |
\includegraphics[scale=0.35]{plots/METsig-RMS.eps} |
| \caption{Plot of RMS as a function the $\not\!\! E_{T}$ significance applied} |
\caption{Plot of RMS as a function the $\not\!\! E_{T}$ significance applied} |
| \label{fig:METsig_RMS} |
\label{fig:METsig_RMS} |
| \end{figure} |
\end{figure} |
| |
|
| |
|
| \clearpage |
%\clearpage |
| |
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|
| In order to check the validity of our emsemble tests procedure, it is instructive to plot both the |
In order to check the validity of our emsemble tests procedure, it is instructive to plot both the |
|
Line 378 initially set in the ensemble and the me
|
Line 386 initially set in the ensemble and the me
|
| curve, that indicates a good behaviour of the fit uncertainties in the ensembles. |
curve, that indicates a good behaviour of the fit uncertainties in the ensembles. |
| |
|
| \begin{figure}[t] |
\begin{figure}[t] |
| \includegraphics[scale=0.5]{plots/gaus_ttbar.eps} |
\includegraphics[scale=0.40]{plots/gaus_ttbar.eps} |
| \caption{Distribution of the output ``measurement'' for an ensemble with 116.9 $\ttbar$ events.} |
\caption{Distribution of the output ``measurement'' for an ensemble with 116.9 $\ttbar$ events.} |
| \label{fig:gaus_ttbar} |
\label{fig:gaus_ttbar} |
| \end{figure} |
\end{figure} |
| |
|
| \begin{figure}[t] |
\begin{figure}[b] |
| \includegraphics[scale=0.5]{plots/pull1-40.eps} |
\includegraphics[scale=0.40]{plots/pull1-40.eps} |
| \caption{The ensemble test's pull.} |
\caption{The ensemble test's pull.} |
| \label{fig:pull} |
\label{fig:pull} |
| \end{figure} |
\end{figure} |
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