| 1 |
The samples used to train the TaNC neural networks are typical of the signals
|
| 2 |
and backgrounds found in common physics analyses using taus. The signal--type
|
| 3 |
training sample is composed of reconstructed tau--candidates that are matched
|
| 4 |
to generator level hadronic tau decays coming from simulated $Z \rightarrow
|
| 5 |
\tau^{+}\tau^{-}$ events. The background training sample consists of
|
| 6 |
reconstructed tau--candidates in simulated QCD $2\rightarrow2$ hard scattering
|
| 7 |
events. The QCD $P_T$ spectrum is steeply falling, and to obtain sufficient
|
| 8 |
statistics across a broad range of $P_T$ the sample is split into different
|
| 9 |
$\hat P_{T}$ bins. Each binned QCD sample imposes a generator level cut on the
|
| 10 |
transverse momentum of the hard interaction. During the evaluation of discrimination
|
| 11 |
performance the QCD samples are weighted according to their respective
|
| 12 |
integrated luminosities to remove any effect of the binning.
|
| 13 |
|
| 14 |
The signal and background samples are split into five subsamples corresponding
|
| 15 |
to each reconstructed decay mode. An additional selection is applied to each
|
| 16 |
subsample by requiring a ``leading pion'': either a charged hadron or gamma
|
| 17 |
candidate with transverse momentum greater than 5 GeV$/c$. A large number of
|
| 18 |
QCD training events is required as both the leading pion selection and the
|
| 19 |
requirement that the decay mode match one of the dominant modes given in table
|
| 20 |
~\ref{tab:decay_modes} are effective discriminants. For each subsample,
|
| 21 |
80\% of the signal and background tau--candidates are used for training the neural
|
| 22 |
networks by the TMVA software, with half (40\%) used as a validation sample
|
| 23 |
used to ensure the neural network is not over--trained. The number of signal and background entries
|
| 24 |
used for training and validation in each decay mode subsample is given in table ~\ref{tab:trainingEvents}.
|
| 25 |
|
| 26 |
%Chained 100 signal files.
|
| 27 |
%Chained 208 background files.
|
| 28 |
%Total signal entries: 874266
|
| 29 |
%Total background entries: 9526176
|
| 30 |
%Pruning non-relevant entries.
|
| 31 |
%After pruning, 584895 signal and 644315 background entries remain.
|
| 32 |
%**********************************************************************************
|
| 33 |
%*********************************** Summary **************************************
|
| 34 |
%**********************************************************************************
|
| 35 |
%* NumEvents with weight > 0 (Total NumEvents) *
|
| 36 |
%*--------------------------------------------------------------------------------*
|
| 37 |
%*shrinkingConePFTauDecayModeProducer ThreeProngNoPiZero: Signal: 53257(53271) Background:155793(155841)
|
| 38 |
%*shrinkingConePFTauDecayModeProducer ThreeProngOnePiZero: Signal: 13340(13342) Background:135871(135942)
|
| 39 |
%*shrinkingConePFTauDecayModeProducer OneProngTwoPiZero: Signal: 34780(34799) Background:51181(51337)
|
| 40 |
%*shrinkingConePFTauDecayModeProducer OneProngOnePiZero: Signal: 136464(138171) Background:137739(139592)
|
| 41 |
%*shrinkingConePFTauDecayModeProducer OneProngNoPiZero: Signal: 300951(345312) Background:144204(161603)
|
| 42 |
|
| 43 |
\begin{table}
|
| 44 |
\centering
|
| 45 |
\begin{tabular}{lcc}
|
| 46 |
%\multirow{2}{*}{} & \multicolumn{2}{c}{Events} \\
|
| 47 |
& Signal & Background \\
|
| 48 |
\hline
|
| 49 |
Total number of tau--candidates & 874266 & 9526176 \\
|
| 50 |
Tau--candidates passing preselection & 584895 & 644315 \\
|
| 51 |
Tau--candidates with $W(P_T,\eta)>0$ & 538792 & 488917 \\
|
| 52 |
\hline
|
| 53 |
Decay Mode & \multicolumn{2}{c}{Training Events} \\
|
| 54 |
\hline
|
| 55 |
$\pi^{-}$ & 300951 & 144204 \\
|
| 56 |
$\pi^{-}\pi^0$ & 135464 & 137739 \\
|
| 57 |
$\pi^{-}\pi^0\pi^0$ & 34780 & 51181 \\
|
| 58 |
$\pi^{-}\pi^{-}\pi^{+}$ & 53247 & 155793 \\
|
| 59 |
$\pi^{-}\pi^{-}\pi^{+}\pi^0$ & 13340 & 135871 \\
|
| 60 |
\end{tabular}
|
| 61 |
\label{tab:trainingEvents}
|
| 62 |
\caption{Number of events used for neural network training and validation for each
|
| 63 |
selected decay mode.}
|
| 64 |
\end{table}
|
| 65 |
|
| 66 |
The remaining 20\% of the signal and background samples are
|
| 67 |
reserved as a statistically independent sample to evaluate the performance of
|
| 68 |
the neural nets after the training is completed. The TaNC uses the ``MLP''
|
| 69 |
neural network implementation provided by the TMVA software package, described
|
| 70 |
in ~\cite{TMVA}. The ``MLP'' classifier is a feed-forward artificial neural
|
| 71 |
network. There are two layers of hidden nodes and a single node in the output
|
| 72 |
layer. The hyperbolic tangent function is used for the neuron activation
|
| 73 |
function.
|
| 74 |
|
| 75 |
The neural networks used in the TaNC have two hidden layers and single node in
|
| 76 |
the output layers. The number of nodes in the first and second hidden layers
|
| 77 |
are chosen to be $N+1$ and $2N+1$, respectively, where $N$ is the number of
|
| 78 |
input observables for that neural network. According to the Kolmogorov's
|
| 79 |
theorem~\cite{Kolmogorov}, any continuous function $g(x)$ defined on a vector
|
| 80 |
space of dimension $d$ spanned by $x$ can be represented by
|
| 81 |
\begin{equation}
|
| 82 |
g(x) = \sum_{j=1}^{j=2d+1} \Phi_j \left(\sum_{i=1}^{d} \phi_i(x) \right)
|
| 83 |
\label{eq:Kolmogorov}
|
| 84 |
\end{equation}
|
| 85 |
for suitably chosen functions for $\Phi_j$ and $\phi_j$. As the form of
|
| 86 |
equation~\ref{eq:Kolmogorov} is similar to the topology of a two hidden--layer
|
| 87 |
neural network, Kolmogorov's theorem suggests that \emph{any} classification
|
| 88 |
problem can be solved with a neural network with two hidden layers containing
|
| 89 |
the appropriate number of nodes.
|
| 90 |
|
| 91 |
The neural network is trained for 500 epochs. At ten epoch intervals, the
|
| 92 |
neural network error is computed using the validation sample to check for
|
| 93 |
over--training (see figure~\ref{fig:overTrainCheck}). The neural network error
|
| 94 |
$E$ is defined~\cite{TMVA} as
|
| 95 |
\begin{equation}
|
| 96 |
E = \frac{1}{2} \sum_{i=1}^N (y_{ANN,i} - \hat y_i)^2
|
| 97 |
\label{eq:NNerrorFunc}
|
| 98 |
%note - not right for weighted dists?
|
| 99 |
\end{equation}
|
| 100 |
where $N$ is the number of training events, $y_{ANN,i}$ is the neural network output
|
| 101 |
for the $i$th training event, and $y_i$ is the desired (-1 for background, 1 for signal) output
|
| 102 |
the $i$th event. No evidence of over--training is observed.
|
| 103 |
|
| 104 |
\begin{figure}[thbp]
|
| 105 |
\setlength{\unitlength}{1mm}
|
| 106 |
\begin{center}
|
| 107 |
\begin{picture}(150, 195)(0,0)
|
| 108 |
\put(0.5, 130)
|
| 109 |
{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngNoPiZero.pdf}}}
|
| 110 |
\put(65, 130)
|
| 111 |
{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngOnePiZero.pdf}}}
|
| 112 |
\put(0.5, 65)
|
| 113 |
{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_OneProngTwoPiZero.pdf}}}
|
| 114 |
\put(65, 65)
|
| 115 |
{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_ThreeProngNoPiZero.pdf}}}
|
| 116 |
\put(33, 0)
|
| 117 |
{\mbox{\includegraphics*[height=60mm]{figures/overtrainCheck_ThreeProngOnePiZero.pdf}}}
|
| 118 |
\end{picture}
|
| 119 |
\caption{
|
| 120 |
Neural network classification error for training (solid red) and testing
|
| 121 |
(dashed blue) samples at ten epoch intervals over the 500 training epochs for each
|
| 122 |
decay mode neural network. The vertical axis represents the classification
|
| 123 |
error, defined by equation~\ref{eq:NNerrorFunc}. N.B. that the choice of
|
| 124 |
hyperbolic tangent for neuron activation functions results in the desired
|
| 125 |
outputs for signal and background to be 1 and -1, respectively. This results
|
| 126 |
in the computed neural network error being larger by a factor of four than
|
| 127 |
the case where the desired outputs are (0, 1). Classifier over--training
|
| 128 |
would be evidenced by divergence of the classification error of the training
|
| 129 |
and testing samples, indicating that the neural net was optimizing about
|
| 130 |
statistical fluctuations in the training sample.
|
| 131 |
}
|
| 132 |
\label{fig:overTrainCheck}
|
| 133 |
\end{center}
|
| 134 |
\end{figure}
|
| 135 |
|
| 136 |
|
| 137 |
The neural networks use as input observables the transverse momentum and $\eta$
|
| 138 |
of the tau--candidates. These observables are included as their correlations
|
| 139 |
with other observables can increase the separation power of the ensemble of
|
| 140 |
observables. For example, the opening angle in $\Delta R$ for signal
|
| 141 |
tau--candidates is inversely related to the transverse momentum, while for
|
| 142 |
background events the correlation is very small~\cite{DavisTau}. In the
|
| 143 |
training signal and background samples, there is significant discrimination
|
| 144 |
power in the $P_T$ spectrum. However, it is desirable to eliminate any
|
| 145 |
systematic dependence of the neural network output on $P_T$ and $\eta$, as in
|
| 146 |
practice the TaNC will be presented with tau--candidates whose $P_T-\eta$
|
| 147 |
spectrum will be analysis dependent. The dependence on $P_T$ and $\eta$ is
|
| 148 |
removed by applying a $P_T$ and $\eta$ dependent weight to the tau--candidates
|
| 149 |
when training the neural nets.
|
| 150 |
|
| 151 |
The weights are defined such that in any region in the vector space spanned by
|
| 152 |
$P_T$ and $\eta$ where the signal sample and background sample probability
|
| 153 |
density functions are different, the sample with higher probability density is
|
| 154 |
weighted such that the samples have identical $P_T-\eta$ probability
|
| 155 |
distributions. This removes regions of $P_T-\eta$ space where the training
|
| 156 |
sample is exclusively signal or background. The weights are computed according to
|
| 157 |
\begin{align*}
|
| 158 |
W(P_T, \eta) &= {\rm less}(p_{sig}(P_T, \eta), p_{bkg}(P_T, \eta))\\
|
| 159 |
w_{sig}(P_T, \eta) &= W(P_T, \eta)/p_{sig}(P_T, \eta) \\
|
| 160 |
w_{bkg}(P_T, \eta) &= W(P_T, \eta)/p_{bkg}(P_T, \eta)
|
| 161 |
\end{align*}
|
| 162 |
where $p_{sig}(P_T,\eta)$ and $p_{bkg}(P_T,\eta)$ are the probability densities of
|
| 163 |
the signal and background samples after the ``leading pion'' and dominant decay mode
|
| 164 |
selections. Figure~\ref{fig:nnTrainingWeights} shows the signal and background
|
| 165 |
training $P_T$ distributions before and after the weighting is applied.
|
| 166 |
|
| 167 |
|
| 168 |
\begin{figure}[thbp]
|
| 169 |
\setlength{\unitlength}{1mm}
|
| 170 |
\begin{center}
|
| 171 |
\begin{picture}(150,60)(0,0)
|
| 172 |
\put(10.5, 2){
|
| 173 |
\mbox{\includegraphics*[height=58mm]{figures/training_weights_unweighted.pdf}}}
|
| 174 |
\put(86.0, 2){
|
| 175 |
\mbox{\includegraphics*[height=58mm]{figures/training_weights_weighted.pdf}}}
|
| 176 |
%\put(-5.5, 112.5){\small (a)}
|
| 177 |
%\put(72.0, 112.5){\small (b)}
|
| 178 |
%\put(-5.5, 54.5){\small (c)}
|
| 179 |
%\put(72.0, 54.5){\small (d)}
|
| 180 |
\end{picture}
|
| 181 |
\caption{Transverse momentum spectrum of signal and background
|
| 182 |
tau--candidates used in neural net training before (left) and after (right) the
|
| 183 |
application of $P_T-\eta$ dependent weight function. Application of the weights
|
| 184 |
lowers the training significance of tau--candidates in regions of $P_T-\eta$
|
| 185 |
phase space where either the signal or background samples has an excess of
|
| 186 |
events. }
|
| 187 |
\label{fig:nnTrainingWeights}
|
| 188 |
\end{center}
|
| 189 |
\end{figure}
|
| 190 |
|
