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friis |
1.1 |
The samples used to train the TaNC neural networks are typical of the signals
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and backgrounds found in common physics analyses using taus. The signal--type
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training sample is composed of reconstucted tau--candidates that are matched to
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generator level hadronic tau decays coming from simulated $Z \rightarrow
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\tau^{+}\tau^{-}$ events. The background training sample consists of reconstructed
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friis |
1.2 |
tau--candidates in simulated QCD $2\rightarrow2$ hard scattering events. The
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QCD $P_T$ spectrum is steeply falling, and to obtain sufficient statistics
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across a broad range of $P_T$ the sample is split into different $\hat P_{T}$
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bins. Each QCD sub-sample imposes a generator level cut on the transverse
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energy of the hard interaction.
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friis |
1.1 |
The signal and background samples are split into five subsamples corresponding
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to each reconstructed decay mode. An additional selection is applied to each
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subsample by requiring a ``leading pion'': either a charged hadron or gamma
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candidate with transverse momentum greater than 5 GeV$/c$. A large number of
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QCD training events is required as the leading pion selection and the
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requirement that the decay mode match one of the dominant modes given in table
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~\ref{tab:decay_modes} are both effective discriminants. For each subsample,
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10000 signal and background tau--candidates are reserved to be used internally
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by the TMVA software to test for over--training. The number of signal and
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background entries used for each decay mode subsample is given in table
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~\ref{tab:trainingEvents}.
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friis |
1.2 |
%Chained 100 signal files.
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%Chained 208 background files.
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%Total signal entries: 874266
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%Total background entries: 9526176
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%Pruning non-relevant entries.
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%After pruning, 584895 signal and 644315 background entries remain.
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%**********************************************************************************
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%*********************************** Summary **************************************
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%**********************************************************************************
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%* NumEvents with weight > 0 (Total NumEvents) *
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%*--------------------------------------------------------------------------------*
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%*shrinkingConePFTauDecayModeProducer ThreeProngNoPiZero: Signal: 53257(53271) Background:155793(155841)
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%*shrinkingConePFTauDecayModeProducer ThreeProngOnePiZero: Signal: 13340(13342) Background:135871(135942)
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%*shrinkingConePFTauDecayModeProducer OneProngTwoPiZero: Signal: 34780(34799) Background:51181(51337)
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%*shrinkingConePFTauDecayModeProducer OneProngOnePiZero: Signal: 136464(138171) Background:137739(139592)
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%*shrinkingConePFTauDecayModeProducer OneProngNoPiZero: Signal: 300951(345312) Background:144204(161603)
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friis |
1.1 |
\begin{table}
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\centering
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friis |
1.2 |
\begin{tabular}{lcc}
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%\multirow{2}{*}{} & \multicolumn{2}{c}{Events} \\
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& Signal & Background \\
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friis |
1.1 |
\hline
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friis |
1.2 |
Total number of tau--candidates & 874266 & 9526176 \\
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Tau--candidates passing preselection & 584895 & 644315 \\
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Tau--candidates with $W(P_T,\eta)>0$ & 538792 & 488917 \\
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friis |
1.1 |
\hline
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friis |
1.2 |
Decay Mode & \multicolumn{2}{c}{Training Events} \\
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friis |
1.1 |
\hline
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friis |
1.2 |
$\pi^{-}$ & 300951 & 144204 \\
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$\pi^{-}\pi^0$ & 135464 & 137739 \\
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$\pi^{-}\pi^0\pi^0$ & 34780 & 51181 \\
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$\pi^{-}\pi^{-}\pi^{+}$ & 53247 & 155793 \\
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$\pi^{-}\pi^{-}\pi^{+}\pi^0$ & 13340 & 135871 \\
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friis |
1.1 |
\end{tabular}
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\label{tab:trainingEvents}
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\caption{Number of events used for neural network training for each
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selected decay mode.}
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\end{table}
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friis |
1.4 |
In both signal and background samples, 20\% of the events are reserved as a
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statistically independent sample to evaluate the performance of the neural nets
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after the training is completed. The TaNC uses the ``MLP'' neural network
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friis |
1.5 |
implementation provided by the TMVA software package, described in
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~\cite{TMVA}. The ``MLP'' classifier is a feed-forward artificial neural
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network. There are two layers of hidden nodes and a single node in
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the output layer. The hyperbolic tangent function is used for the neuron activation function.
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The number of hidden nodes in the first and second layers
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are determined by Kolmogorov's theorem~\cite{kolmogorovsTheorem}; the number of
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hidden nodes in the first (second) layer is $N+1 (2*N+1)$, where $N$ is the
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number of input observables. The neural network is trained for 500 epochs. At
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ten epoch intervals, the neural network error is computed to check for
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overtraining. (see figure~\ref{fig:overTrainCheck}) The neural network error $E$ is
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defined~\cite{TMVA} as
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\begin{equation}
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E = \frac{1}{2} \sum_{i=1}^N (y_{ANN,i} - \hat y_i)^2
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\label{eq:NNerrorFunc}
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%note - not right for weighted dists?
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\end{equation}
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where $N$ is the number of training events, $y_{ANN,i}$ is the neural network output
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for the $i$th training event, and $y_i$ is the desired (-1 for background, 1 for signal) output
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the $i$th event. No evidence of overtraining is observed.
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friis |
1.4 |
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\begin{figure}[t]
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\setlength{\unitlength}{1mm}
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\begin{center}
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\begin{picture}(150, 195)(0,0)
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\put(0.5, 130)
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{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngNoPiZero.pdf}}}
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\put(65, 130)
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{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngOnePiZero.pdf}}}
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\put(0.5, 65)
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{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngTwoPiZero.pdf}}}
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\put(65, 65)
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{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_ThreeProngNoPiZero.pdf}}}
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\put(33, 0)
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{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_ThreeProngOnePiZero.pdf}}}
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\end{picture}
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friis |
1.5 |
\caption{
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Neural network classification error for training (blue) and testing
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(red) samples at ten epoch intervals over the 500 training epochs for each
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decay mode neural network. The vertical axis represents the classification
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error, defined by equation~\ref{eq:NNerrorFunc}. N.B. that the choice of
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hyperbolic tangent for neuron activation functions results in the desired
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outputs for signal and backround to be -1 and 1, respectively. This results
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in the computed neural network error being larger by a factor of four than
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the case where the desired outputs are (0, 1). Classifier over--training
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would be evidenced by divergence of the classification error of the training
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and testing samples, indicating that the neural net was optimizing about
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statistical fluctations in the training sample.
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friis |
1.4 |
}
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\label{fig:overTrainCheck}
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\end{center}
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\end{figure}
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friis |
1.1 |
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friis |
1.2 |
The neural nets uses as input variables the transverse momentum and $\eta$ of the
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friis |
1.1 |
tau--candidates. These variables are included as their correlations with other
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observables can increase the separation power of the ensemble of observables.
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For example, the opening angle in $\Delta R$ for signal tau--candidates is
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inversely related to the transverse momentum, while for background events the
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correlation is very small (see~\cite{DavisTau}). In the training signal and
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background samples, there is significant discrimination power in the $P_T$
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spectrum. However, it is desirable to eliminate any systematic dependence of
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friis |
1.2 |
the neural network output on $P_T$ and $\eta$, as in use the TaNC will be
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presented with tau--candidates whose $P_T-\eta$ spectrum will be analysis
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dependent. The dependence on $P_T$ and $\eta$ is removed by applying a $P_T$ and
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$\eta$ dependent weight to the tau--candidates when training the neural nets.
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The weights are defined such that in any region in $P_T-\eta$ where the signal
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and background probability density function are different, the sample with
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higher probability density is weighted such that the samples have identical
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$P_T-\eta$ probability distributions. This removes regions of $P_T-\eta$ space
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where the training sample is exclusively signal or background. The weights are
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computed by
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\begin{align*}
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W(P_T, \eta) &= {\rm less}(p_{sig}(P_T, \eta), p_{bkg}(P_T, \eta))\\
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w_{sig}(P_T, \eta) &= W(P_T, \eta)/p_{sig}(P_T, \eta) \\
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w_{bkg}(P_T, \eta) &= W(P_T, \eta)/p_{bkg}(P_T, \eta)
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\end{align*}
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where $p_{sig}(P_T,\eta)$ and $p_{bkg}(P_T,\eta)$ are the probility densities of
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the signal and background samples after the ``leading pion'' and decay mode
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selections. Figure~\ref{fig:nnTrainingWeights} shows the signal and background
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training $P_T$ distributions before and after the weighting is applied.
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\begin{figure}[t]
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\setlength{\unitlength}{1mm}
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\begin{center}
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\begin{picture}(150,60)(0,0)
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\put(10.5, 2){
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\mbox{\includegraphics*[height=58mm]{figures/training_weights_unweighted.pdf}}}
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\put(86.0, 2){
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\mbox{\includegraphics*[height=58mm]{figures/training_weights_weighted.pdf}}}
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%\put(-5.5, 112.5){\small (a)}
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%\put(72.0, 112.5){\small (b)}
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%\put(-5.5, 54.5){\small (c)}
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%\put(72.0, 54.5){\small (d)}
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\end{picture}
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friis |
1.4 |
\caption{Transverse momentum spectrum of signal and background
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| 164 |
friis |
1.2 |
tau--candidates used in neural net training before (left) and after (right) the
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application of $P_T-\eta$ dependent weight function. Application of the weights
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lowers the training significance of tau--candidates in regions of $P_T-\eta$
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phase space where either the signal or background samples has an excess of
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events. }
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\label{fig:nnTrainingWeights}
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\end{center}
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\end{figure}
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