Annotation of ttbar/p20_taujets_note/NN.tex, revision 1.1
1.1 ! uid12904 1: \newpage
! 2:
! 3: \section{\label{sub:NN}Neural Network Analysis}
! 4:
! 5: \subsection{\label{sub:Variables}Variables for NN training}
! 6:
! 7: \noindent Following the same procedure as in the previous analysis, we determine the content
! 8: of signal and background in the preselected sample, increase signal/background rate
! 9: and from this, measure the cross-section.
! 10: The procedure adopted in the p17 analysis was feed a set of topological variables into an
! 11: artificial neural network in order to provide the best possible separation between
! 12: signal and background. As before, the criteria for choosing such variables were: power of discrimination
! 13: and $\tau$-uncorrelated variables. The set is presented below:
! 14:
! 15: \begin{itemize}
! 16: \item \textit{\textbf{$H_{T}$}} - the scalar sum of all jet's $p_{T}$ (here and below including $\tau$ lepton candidates).
! 17:
! 18: \item \textit{\textbf{$\not\!\! E_{T}$ significance}} - As being the variable that provides the best signal-background
! 19: separation we decided to optimize it.
! 20:
! 21: \item \textit{\textbf{Aplanarity}} \cite{p17topo} - the normalized momentum tensor is defined as
! 22:
! 23: \begin{center}
! 24: \begin{equation}
! 25: {\cal M} = \frac{\sum_{o}p^{o}_{i}p^{o}_{j}}{\sum_{o}|\overrightarrow{p^{o}}|}
! 26: \label{tensor}
! 27: \end{equation}
! 28: \end{center}
! 29:
! 30: \noindent where $\overrightarrow{p^{0}}$ is the momentum-vector of a reconstructed object $o$
! 31: and $i$ and $j$ are cartesian coordinates. From the diagonalization of $\cal M$ we find three eigenvalues
! 32: $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ with the constraint $\lambda_{1} + \lambda_{2} + \lambda_{3} = 1$.
! 33: The aplanarity {\cal A} is given by {$\cal A$} = $\frac{3}{2}\lambda_{3}$ and measures the flatness of an event.
! 34: Hence, it is defined in the range $0 \leq {\cal M} \leq 0.5$. Large values of {$\cal A$} correspond to more spherical events,
! 35: like $t\bar{t}$ events for instance, since they are typical of decays of heavy objects. On the other hand,
! 36: both QCD and $W + \mbox{jets}$ events are more planar since jets in these events are primarily due to
! 37: initial state radiation.
! 38:
! 39: \item \textit{\textbf{Sphericity}} \cite{p17topo} - being defined as {$\cal S$} = $\frac{3}{2}(\lambda_{2} + \lambda_{3})$,
! 40: and having a range $0 \leq {\cal S} \leq 1.0$, sphericity is a measure of the summed $p^{2}_{\perp}$ with
! 41: respect to the event axis. In this sense a 2-jets event corresponds to {$\cal S$} $\approx 0$ and an isotropic event
! 42: {$\cal S$} $\approx 1$. $t\bar{t}$ events are very isotropic as they are typical of the decays of heavy objects
! 43: and both QCD and $W + \mbox{jets}$ events are less isotropic due to the fact that jets in these events come
! 44: primarily from initial state radiation.
! 45:
! 46: \item \textit{\textbf{Top and $W$ mass likelihood}} - a $\chi^{2}$-like variable.
! 47: $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma_{t}}\right)^{2}+\left(\frac{M_{2j}-M_{W}}{\sigma_{W}}\right)^{2}$,
! 48: where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (172.4
! 49: GeV and 81.02 GeV respectively) and resolution values (19.4 GeV and 8.28
! 50: GeV respectively). $M_{3j}$ and $M_{2j}$ are invariant masses composed
! 51: of the jet combinations. We choose combination that minimizes $L$.
! 52:
! 53: \item \textit{\textbf{Centrality}}, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum
! 54: of energies of the jets.
! 55: \item \textit{\textbf{$\cos(\theta*)$}} - The angle between the beam axis and the
! 56: highest-$p_T$ jet in the rest frame of all the jets in the event.
! 57: \item \textit{\textbf{$\sqrt(s)$}} - The invariant mass of all jets and $\tau$s in the event.
! 58:
! 59: \end{itemize}
! 60:
! 61: The chosen variables are in the end a consequence of the method employed in this
! 62: analysis: use events from the QCD-enriched
! 63: loose-tight sample to model QCD events in the signal-rich sample, and use
! 64: a b-tag veto sample as an independent control sample to check the validity of such
! 65: background modeling.
! 66:
! 67: %\clearpage
! 68:
! 69: \subsection{\label{sub:NN-variables}Topological NN}
! 70: For training the Neural Network we used the Multilayer Perceptron algorithm, as described in
! 71: \cite{MLPfit}. As explained before in Section \ref{sub:Results-of-the}, the first 1400000
! 72: events in the ``loose-tight'' sample were used as background
! 73: for NN training for taus types 1 and 2, and the first 600000 of the same sample for NN training for type 3 taus.
! 74: This means that different tau types are being treated separately in the topological NN.
! 75: 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
! 76: for the measurement.
! 77: When doing the measurement later on (Section \ref{sub:xsect}) we pick the tau with the highest $NN(\tau)$
! 78: in the signal sample as the tau cadidate at same time that taus in the loose-tight sample are picked at
! 79: random since all of them are regarded as fake taus by being below the cut $NN(\tau)$ = 0.7. By doing this
! 80: we expect to avoid any bias when selecting real taus for the measurement.
! 81: Figures \ref{fig:nnout_type2_training} and \ref{fig:nnout_type3_training} show the
! 82: effect of each of the chosen the topological event NN input variables on the final output.
! 83:
! 84: Figures \ref{fig:nnout_type2} and \ref{fig:nnout_type3} show the NN output as a result of the training
! 85: described above. It is evident from both pictures that high values of NN correspond to
! 86: the signal-enriched region.
! 87:
! 88:
! 89:
! 90: \begin{figure}[h]
! 91: \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type2_tauQCD.eps}
! 92: \caption{Training of topological Neural Network output for type 1 and 2 $\tau$ channel.
! 93: Upper left: relative impact of each of the input variables; upper right: topological structure;
! 94: lower right: final signal-background separation of the method; lower left: convergence curves.}
! 95: \label{fig:nnout_type2_training}
! 96: \end{figure}
! 97:
! 98: \newpage
! 99:
! 100:
! 101: \begin{figure}[h]
! 102: \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type3_tauQCD.eps}
! 103: \caption{Training of topological Neural Network output for type 3 $\tau$ channel.
! 104: Upper left: relative impact of each of the input variables; upper right: topological structure;
! 105: lower right: final signal-background separation of the method; lower left: convergence curves.}
! 106: \label{fig:nnout_type3_training}
! 107: \end{figure}
! 108:
! 109: \begin{figure}[h]
! 110: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeI_II/nnout.eps}
! 111: \caption{The topological Neural Network output for type 1 and 2 $\tau$ channel}
! 112: \label{fig:nnout_type2}
! 113: \end{figure}
! 114:
! 115: \newpage
! 116:
! 117: \begin{figure}[t]
! 118: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeIII/nnout.eps}
! 119: \caption{The topological Neural Network output for type 3 $\tau$ channel}
! 120: \label{fig:nnout_type3}
! 121: \end{figure}
! 122:
! 123:
! 124: \subsection{\label{sub:NN-optimization}NN optimization}
! 125: One difference between this present analysis and the previous p17 is that we performed a NN optimization along with a
! 126: $\not\!\! E_{T}$ significance optimization. Previously a cut of $>$ 3.0 was applied to $\not\!\! E_{T}$ significance
! 127: at the preselection stage and then it was included as one of the variables for NN training. This time as we
! 128: chose to optimize it, since it is still a good variable to provide signal-background discrimination (Figure \ref{fig:metl_note}).
! 129: It is important to stress out that after the optimization we performed the
! 130: analysis with the optimized $\not\!\! E_{T}$ significance cut
! 131: applied when doing both $\tau$ and b ID (Section \ref{sub:Results-of-the}), therefore
! 132: after the preselection where no $\not\!\! E_{T}$ significance cut was applied.
! 133: We then went back anp reprocessed (preselected) all MC samples with the optimized cut. Both results,
! 134: with $\not\!\! E_{T}$ significance applied during and after preselectio were identical.
! 135: We then chose to present this analyis with this cut applied at the preselection
! 136: level in order to have a consistent cut flow throughout the analysis(Section \ref{sub:Preselection}).
! 137:
! 138:
! 139: \begin{figure}[h]
! 140: \includegraphics[scale=0.5]{plots/metl_allEW.eps}
! 141: \caption{$\not\!\! E_{T}$ significance distribution for signal and backgrounds.}
! 142: \label{fig:metl_note}
! 143: \end{figure}
! 144:
! 145:
! 146: \newpage
! 147:
! 148: Below we describe how we split this part of the analysis into two parts:
! 149:
! 150: \begin{enumerate}
! 151: \item {\bf Set optimization:} We applied an arbitrary cut on $\not\!\! E_{T}$ significance of $\geq$ 4.0 and
! 152: varied the set of varibles going into NN training
! 153: \item {\bf $\not\!\! E_{T}$ significance optimization:} After chosing the best set based on the lowest RMS,
! 154: we then varied the $\not\!\! E_{T}$ significance cut
! 155: \end{enumerate}
! 156:
! 157: For this part of the analysis we present the sets of variables that were taken into account to perform the NN traning
! 158: \begin{itemize}
! 159: \item \textit{\textbf{Set I}} : {$H_{T}$}, aplan (aplanarity), sqrts ($\sqrt{s}$)
! 160: \item \textit{\textbf{Set II}} : {$H_{T}$}, aplan, cent (centrality)
! 161: \item \textit{\textbf{Set III}} : {$H_{T}$}, aplan, spher (spherecity)
! 162: \item \textit{\textbf{Set IV}} : {$H_{T}$}, cent, spher
! 163: \item \textit{\textbf{Set V}} : aplan, cent, spher
! 164: \item \textit{\textbf{Set VI}} : {$H_{T}$}, aplan, sqrts, spher
! 165: \item \textit{\textbf{Set VII}} : {$H_{T}$}, aplan, sqrts, cent
! 166: \item \textit{\textbf{Set VIII}} : {$H_{T}$}, aplan, sqrts, costhetastar ($cos(\theta^{*})$
! 167: \item \textit{\textbf{Set IX}} : {$H_{T}$}, aplan, sqrts, cent, spher
! 168: \item \textit{\textbf{Set X}} : {$H_{T}$}, aplan, sqrts, cent, costhetastar
! 169: \item \textit{\textbf{Set XI}} : {$H_{T}$}, aplan, sqrts, spher, costhetastar
! 170: \item \textit{\textbf{Set XII}} : metl, {$H_{T}$}, aplan, sqrts
! 171: \item \textit{\textbf{Set XIII}} : metl, {$H_{T}$}, aplan, cent
! 172: \item \textit{\textbf{Set XIV}} : metl, {$H_{T}$}, aplan, spher
! 173: \item \textit{\textbf{Set XV}} : metl, {$H_{T}$}, cent, spher
! 174: \item \textit{\textbf{Set XVI}} : metl, {$H_{T}$}, aplan
! 175: \item \textit{\textbf{Set XVII}} : metl, {$H_{T}$}, sqrts
! 176: \item \textit{\textbf{Set XVIII}} : metl, aplan, sqrts
! 177: \item \textit{\textbf{Set XIX}} : metl, {$H_{T}$}, cent
! 178: \item \textit{\textbf{Set XX}} : metl, {$H_{T}$}, aplan, sqrts, cent
! 179: \item \textit{\textbf{Set XXI}} : metl, {$H_{T}$}, aplan, cent, spher
! 180: \item \textit{\textbf{Set XXII}} : metl, {$H_{T}$}, aplan, sqrts, spher
! 181: \item \textit{\textbf{Set XXIII}} : metl, {$H_{T}$}, aplan, sqrts, costhetastar
! 182: \item \textit{\textbf{Set XXIV}} : metl, sqrts, cent, spher, costhetastar
! 183: \item \textit{\textbf{Set XXV}} : metl, {$H_{T}$}, cent, spher, costhetastar
! 184: \item \textit{\textbf{Set XXVI}} : metl, aplan, cent, spher, costhetastar
! 185: \item \textit{\textbf{Set XXVII}} : metl, {$H_{T}$}, aplan, cent, costhetastar
! 186: \item \textit{\textbf{Set XXVIII}} : {$H_{T}$}, aplan, topmassl
! 187: \item \textit{\textbf{Set XXIX}} : {$H_{T}$}, aplan, sqrts, topmassl
! 188: \item \textit{\textbf{Set XXX}} : {$H_{T}$}, aplan, sqrts, cent, topmassl
! 189: \item \textit{\textbf{Set XXXI}} : {$H_{T}$}, aplan, sqrts, costhetastar, topmassl
! 190: \item \textit{\textbf{Set XXXII}} : metl, {$H_{T}$}, topmassl, aplan, sqrts
! 191: \item \textit{\textbf{Set XXXIII}} : metl, spher, costhetastar, aplan, cent
! 192: % \item \textit{\textbf{Set XXXIV}} : metl, spher, sqrts, topmassl, ktminp
! 193: \end{itemize}
! 194:
! 195: The criteria used for making a decision on which variable should be used follow:
! 196: \begin{itemize}
! 197: \item No more than 5 variables to keep NN simple and stable. More variables leads to instabilities (different
! 198: result after each retraining) and require larger training samples.
! 199: % \item We want to use $metl$ (\met significance) variable, since it's the one providing best discrimination.
! 200: \item We do not want to use highly correlated variables in same NN.
! 201: % \item We can not use tau-based variables.
! 202: \item We want to use variables with high discriminating power.
! 203: \end{itemize}
! 204:
! 205: In order to make the decision about which of these 11 choices is the optimal we created an ensemble of
! 206: 20000 pseudo-datasets each containing events randomly (according to a Poisson distribution) picked
! 207: from QCD, EW and $\ttbar$ templates. Each of these datasets was treated like real data, meaning applying all
! 208: the cuts and doing the shape fit of event topological NN. QCD templates for fit were made from the same
! 209: ``loose-tight $\tau$ sample'' from which the QCD component of the ``data'' was drawn.
! 210: The figure of merit chosen is given by Equation \ref{merit} below:
! 211:
! 212: \begin{equation}
! 213: f = \displaystyle \frac{(N_{fit} - N_{true})}{N_{true}}
! 214: \label{merit}
! 215: \end{equation}
! 216:
! 217:
! 218: \noindent where $N_{fit}$ is the number of $t\bar{t}$ pairs given by
! 219: the fit and $N_{true}$ is the number of $t\bar{t}$ pairs from the Poisson distribution.
! 220: In both Set and $\not\!\! E_{T}$ significance optimization, the lowest RMS was used
! 221: to caracterize which configuration is the best in each case.
! 222:
! 223:
! 224: The plots showing results concerning the set optimizations are found in Appendix \ref{app:set_opt} and are summarized
! 225: in Table \ref{setopt_table} below, where each RMS and mean are shown.
! 226: For NN training is standard to choose the number of hidden nodes as being twice
! 227: the number the number of variables used for the training. The parenthesis after each set ID show the number of
! 228: hidden nodes in NN training.
! 229:
! 230: \begin{table}[htbp]
! 231: \begin{tabular}{|c|r|r|r|} \hline
! 232: Set of variables & \multicolumn{1}{c|}{RMS} & \multicolumn{1}{c|}{mean} \\ \hline
! 233:
! 234: \hline
! 235:
! 236:
! 237: Set1(6) & \multicolumn{1}{c|}{0.1642} & \multicolumn{1}{c|}{0.0265}\\ \hline
! 238:
! 239: Set2(6) & \multicolumn{1}{c|}{0.1840} & \multicolumn{1}{c|}{0.0054}\\ \hline
! 240:
! 241: Set3(6) & \multicolumn{1}{c|}{0.1923} & \multicolumn{1}{c|}{0.0060}\\ \hline
! 242:
! 243: Set4(6) & \multicolumn{1}{c|}{0.1978} & \multicolumn{1}{c|}{0.0175}\\ \hline
! 244:
! 245: Set5(6) & \multicolumn{1}{c|}{0.2385} & \multicolumn{1}{c|}{0.0022}\\ \hline
! 246:
! 247: Set6(8) & \multicolumn{1}{c|}{0.1687} & \multicolumn{1}{c|}{0.0115}\\ \hline
! 248:
! 249: Set7(8) & \multicolumn{1}{c|}{0.1667} & \multicolumn{1}{c|}{0.0134}\\ \hline
! 250:
! 251: Set8(10) & \multicolumn{1}{c|}{0.1668} & \multicolumn{1}{c|}{0.0162}\\ \hline
! 252:
! 253: Set9(10) & \multicolumn{1}{c|}{0.1721} & \multicolumn{1}{c|}{0.0102}\\ \hline
! 254:
! 255: Set10(10) & \multicolumn{1}{c|}{0.1722} & \multicolumn{1}{c|}{0.0210}\\ \hline
! 256:
! 257: Se11(10) & \multicolumn{1}{c|}{0.1716} & \multicolumn{1}{c|}{0.0180}\\ \hline
! 258:
! 259: Set12(8) & \multicolumn{1}{c|}{0.1662} & \multicolumn{1}{c|}{0.0039}\\ \hline
! 260:
! 261: Set13(8) & \multicolumn{1}{c|}{0.1819} & \multicolumn{1}{c|}{0.0018}\\ \hline
! 262:
! 263: Set14(8) & \multicolumn{1}{c|}{0.1879} & \multicolumn{1}{c|}{0.0019}\\ \hline
! 264:
! 265: Set15(8) & \multicolumn{1}{c|}{0.1884} & \multicolumn{1}{c|}{-0.0004}\\ \hline
! 266:
! 267: Set16(6) & \multicolumn{1}{c|}{0.1912} & \multicolumn{1}{c|}{0.0034}\\ \hline
! 268:
! 269: Set17(6) & \multicolumn{1}{c|}{0.1768} & \multicolumn{1}{c|}{0.0074}\\ \hline
! 270:
! 271: Set18(6) & \multicolumn{1}{c|}{0.2216} & \multicolumn{1}{c|}{-0.0030}\\ \hline
! 272:
! 273: Set19(6) & \multicolumn{1}{c|}{0.1921} & \multicolumn{1}{c|}{0.0015}\\ \hline
! 274:
! 275: Set20(10) & \multicolumn{1}{c|}{0.1620} & \multicolumn{1}{c|}{0.0262}\\ \hline
! 276:
! 277: Set21(10) & \multicolumn{1}{c|}{0.1753} & \multicolumn{1}{c|}{0.0010}\\ \hline
! 278:
! 279: Set22(10) & \multicolumn{1}{c|}{0.1646} & \multicolumn{1}{c|}{0.0086}\\ \hline
! 280:
! 281: Set23(10) & \multicolumn{1}{c|}{0.1683} & \multicolumn{1}{c|}{0.0132}\\ \hline
! 282:
! 283: Set24(10) & \multicolumn{1}{c|}{0.2053} & \multicolumn{1}{c|}{0.0122}\\ \hline
! 284:
! 285: Set25(10) & \multicolumn{1}{c|}{0.1906} & \multicolumn{1}{c|}{0.0038}\\ \hline
! 286:
! 287: Set26(10) & \multicolumn{1}{c|}{0.2130} & \multicolumn{1}{c|}{0.0028}\\ \hline
! 288:
! 289: Set27(10) & \multicolumn{1}{c|}{0.1859} & \multicolumn{1}{c|}{0.0004}\\ \hline
! 290:
! 291: Set28(6) & \multicolumn{1}{c|}{0.1910} & \multicolumn{1}{c|}{-0.0022}\\ \hline
! 292:
! 293: Set29(8) & \multicolumn{1}{c|}{0.1587} & \multicolumn{1}{c|}{0.0214}\\ \hline
! 294:
! 295: Set30(10) & \multicolumn{1}{c|}{0.1546} & \multicolumn{1}{c|}{0.0148}\\ \hline
! 296:
! 297: Set31(10) & \multicolumn{1}{c|}{0.1543} & \multicolumn{1}{c|}{0.0203}\\ \hline
! 298:
! 299: Set32(10) & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
! 300:
! 301: Set33(10) & \multicolumn{1}{c|}{0.2201} & \multicolumn{1}{c|}{0.0081}\\ \hline
! 302:
! 303: %Set34(10) & \multicolumn{1}{c|}{0.1955} & \multicolumn{1}{c|}{0.0184}\\ \hline
! 304: \end{tabular}
! 305: \caption{Results for set optimization part whit $\not\!\! E_{T}$ significance $>$ 4.0 applied to all sets.
! 306: The number in parenthesis refers to number of hidden nodes in each case.}
! 307: \label{setopt_table}
! 308: \end{table}
! 309:
! 310: From Table \ref{setopt_table} we see that Set I has the lowest RMS, thus we chose it
! 311: as the set to be used in $\not\!\! E_{T}$ significance optimization part, whose results are
! 312: shown in Appendix \ref{app:metl_opt} and then summarized in Table \ref{metlopt_table} below
! 313:
! 314: \begin{table}[htbp]
! 315: \begin{tabular}{|c|r|r|r|} \hline
! 316: Set of variables & $\not\!\! E_{T}$ significance cut & RMS & \multicolumn{1}{c|}{mean} \\ \hline
! 317:
! 318: \hline
! 319:
! 320:
! 321: %Set6(10) & \multicolumn{1}{c|}{1.0} & \multicolumn{1}{c|}{0.2611} \\ \hline
! 322:
! 323: %Set6(10) & \multicolumn{1}{c|}{1.5} & \multicolumn{1}{c|}{0.2320} \\ \hline
! 324:
! 325: %Set6(10) & \multicolumn{1}{c|}{2.0} & \multicolumn{1}{c|}{0.2102} \\ \hline
! 326:
! 327: %Set6(10) & \multicolumn{1}{c|}{2.5} & \multicolumn{1}{c|}{0.2021} \\ \hline
! 328:
! 329: Set32(10) & \multicolumn{1}{c|}{3.0} & \multicolumn{1}{c|}{0.1507} & \multicolumn{1}{c|}{0.0157}\\ \hline
! 330:
! 331: Set32(10) & \multicolumn{1}{c|}{3.5} & \multicolumn{1}{c|}{0.1559} & \multicolumn{1}{c|}{0.0189}\\ \hline
! 332:
! 333: Set32(10) & \multicolumn{1}{c|}{4.0} & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
! 334:
! 335: Set32(10) & \multicolumn{1}{c|}{4.5} & \multicolumn{1}{c|}{0.1511} & \multicolumn{1}{c|}{0.0153}\\ \hline
! 336:
! 337: Set32(10) & \multicolumn{1}{c|}{5.0} & \multicolumn{1}{c|}{0.1552} & \multicolumn{1}{c|}{0.0205}\\ \hline
! 338:
! 339: %Set6(10) & \multicolumn{1}{c|}{5.5} & \multicolumn{1}{c|}{0.4008} \\ \hline
! 340: \end{tabular}
! 341: \caption{Results for $\not\!\! E_{T}$ significance optimization part when varying the $\not\!\! E_{T}$ significance cut
! 342: The number in parenthesis refers to number of hidden nodes in each case.}
! 343: \label{metlopt_table}
! 344: \end{table}
! 345:
! 346:
! 347:
! 348: Combined results from Tables \ref{setopt_table} and \ref{metlopt_table} show that the best configuration found
! 349: was Set I with $\not\!\! E_{T}$ significance $\geq$ 4.0. Therefore, this was the
! 350: configuration used to perform the cross-section measurement.Figure \ref{fig:METsig_RMS} shows the variation of the RMS as function
! 351: of the $\not\!\! E_{T}$ significance we applied.
! 352:
! 353: \begin{figure}[b]
! 354: \includegraphics[scale=0.4]{plots/METsig-RMS.eps}
! 355: \caption{Plot of RMS as a function the $\not\!\! E_{T}$ significance applied}
! 356: \label{fig:METsig_RMS}
! 357: \end{figure}
! 358:
! 359:
! 360: \clearpage
! 361:
! 362:
! 363: In order to check the validity of our emsemble tests procedure, it is instructive to plot both the
! 364: distribution of the predicted number of $t\bar{t}$ and what is called ``pull'', defined in Equation
! 365: \ref{pull} below:
! 366:
! 367: \begin{equation}
! 368: p = \displaystyle \frac{(N_{fit}-N_{true})}{\sigma_{fit}}
! 369: \label{pull}
! 370: \end{equation}
! 371:
! 372: \noindent where $\sigma_{fit}$ is the error on the number of $t\bar{t}$ pairs given by the fit.
! 373:
! 374: Figures \ref{fig:gaus_ttbar} and \ref{fig:pull} show both beforementioned distributions.
! 375:
! 376: From Figure \ref{fig:gaus_ttbar} we see a good agreement between the number of $t\bar{t}$ pairs
! 377: initially set in the ensemble and the measured value. And Figure \ref{fig:pull} shows a nice gaussian
! 378: curve, that indicates a good behaviour of the fit uncertainties in the ensembles.
! 379:
! 380: \begin{figure}[t]
! 381: \includegraphics[scale=0.5]{plots/gaus_ttbar.eps}
! 382: \caption{Distribution of the output ``measurement'' for an ensemble with 116.9 $\ttbar$ events.}
! 383: \label{fig:gaus_ttbar}
! 384: \end{figure}
! 385:
! 386: \begin{figure}[t]
! 387: \includegraphics[scale=0.5]{plots/pull1-40.eps}
! 388: \caption{The ensemble test's pull.}
! 389: \label{fig:pull}
! 390: \end{figure}
! 391:
! 392:
! 393: %\newpage
! 394:
! 395:
! 396:
! 397:
! 398:
! 399: \clearpage
! 400:
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