Annotation of ttbar/p20_taujets_note/NN.tex, revision 1.3
1.1 uid12904 1: \newpage
1.3 ! uid12904 2:
1.1 uid12904 3: \section{\label{sub:NN}Neural Network Analysis}
4:
5: \subsection{\label{sub:Variables}Variables for NN training}
6:
1.3 ! uid12904 7: \noindent Following the same procedure as in the p17 analysis, we determine the content
1.1 uid12904 8: of signal and background in the preselected sample, increase signal/background rate
1.3 ! uid12904 9: and from this, measure the cross section.
! 10: In p17, an artificil neural network based on topological characteristics of an event was used to
! 11: extract signal from a background-enriched region. As before, the criteria used in choosing the variables were:
! 12: power of discrimination and $\tau$-uncorrelated variables. The following variables were considered:
1.1 uid12904 13:
14: \begin{itemize}
1.3 ! uid12904 15: \item \textit{\textbf{$H_{T}$}} - The scalar sum of all jet $p_{T}$'s (here and below including $\tau$ lepton candidates).
! 16: For $H_{T}$ values above $\sim$ 200 GeV we observed a dominance of signal over background.
1.1 uid12904 17:
1.3 ! uid12904 18: \item \textit{\textbf{$\not\!\! E_{T}$ significance}} - It is computed from calculated resolutions of
! 19: physical objects (jets, electrons, muons and unclustered energy) \cite{p17_note,METsig}.
! 20: It was chosen to be used and optimized due to its good signal-background discrimination power.
1.1 uid12904 21:
22: \item \textit{\textbf{Aplanarity}} \cite{p17topo} - the normalized momentum tensor is defined as
23:
24: \begin{center}
25: \begin{equation}
1.3 ! uid12904 26: {\cal M}_{ab} \equiv \frac{\sum_{i}p_{ia}p_{ib}}{\sum_{i}p^{2}_{i}}
1.1 uid12904 27: \label{tensor}
28: \end{equation}
29: \end{center}
30:
1.3 ! uid12904 31: \noindent where $p_{i}$ is the momentum-vector
! 32: and teh index $i$ runs over all the jets and the $W$. From the diagonalization of $\cal M$ we find three eigenvalues
1.1 uid12904 33: $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ with the constraint $\lambda_{1} + \lambda_{2} + \lambda_{3} = 1$.
1.3 ! uid12904 34: The aplanarity is defined as {$\cal A$} = $\frac{3}{2}\lambda_{3}$ and measures the flatness of an event.
! 35: It assumes values in the range $0 \leq {\cal A} \leq 0.5$.
! 36: It was chosen to be used in the NN due to the fact that large values of {$\cal A$} correspond to more spherical events,
! 37: like $t\bar{t}$ events for instance, since they are typical of cascade decays of heavy objects. On the other hand,
! 38: both QCD and $W + \mbox{jets}$ events tend to be more collinear since jets in these events are primarily due to
1.1 uid12904 39: initial state radiation.
40:
1.3 ! uid12904 41: \item \textit{\textbf{Sphericity}} \cite{p17topo} - Defined as {$\cal S$} = $\frac{3}{2}(\lambda_{2} + \lambda_{3})$,
! 42: and ranges as $0 \leq {\cal S} \leq 1.0$, sphericity is a measure of the summed $p^{2}_{\perp}$
! 43: More isotropic events have {$\cal S$} $\approx 1$ while less isotropic ones have {$\cal S$} $\approx 0$.
! 44: Sphericity is a good discrminator since $t\bar{t}$ events are very isotropic as they are typical of the decays of heavy objects
1.1 uid12904 45: and both QCD and $W + \mbox{jets}$ events are less isotropic due to the fact that jets in these events come
46: primarily from initial state radiation.
47:
48: \item \textit{\textbf{Top and $W$ mass likelihood}} - a $\chi^{2}$-like variable.
49: $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma_{t}}\right)^{2}+\left(\frac{M_{2j}-M_{W}}{\sigma_{W}}\right)^{2}$,
50: where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (172.4
51: GeV and 81.02 GeV respectively) and resolution values (19.4 GeV and 8.28
52: GeV respectively). $M_{3j}$ and $M_{2j}$ are invariant masses composed
1.3 ! uid12904 53: of 2- and 3-jet combinations. We choose combination that minimizes $L$.
1.1 uid12904 54:
55: \item \textit{\textbf{Centrality}}, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum
1.3 ! uid12904 56: of energies of the jets. Used as discrimination variable since highe values ($\sim$ 1.0) are
! 57: more signal-dominated while low values ($\sim$ 0) are more background-dominated.
! 58:
1.1 uid12904 59: \item \textit{\textbf{$\cos(\theta*)$}} - The angle between the beam axis and the
1.3 ! uid12904 60: highest-$p_T$ jet in the rest frame of all the jets in the event. $t\bar{t}$ events tend to have
! 61: a lower ($\sim$ 0) $\cos(\theta*)$ values. This motivated its choice.
! 62:
! 63: \item \textit{\textbf{$M_{jj\tau}$}} - The invariant mass of all jets and $\tau$s in the event.
1.1 uid12904 64:
65: \end{itemize}
66:
67: The chosen variables are in the end a consequence of the method employed in this
68: analysis: use events from the QCD-enriched
69: loose-tight sample to model QCD events in the signal-rich sample, and use
70: a b-tag veto sample as an independent control sample to check the validity of such
1.3 ! uid12904 71: background modeling. Plots of all variables described above are found in Appendix \ref{app:discri_var}.
1.1 uid12904 72:
73: %\clearpage
74:
75: \subsection{\label{sub:NN-variables}Topological NN}
76: For training the Neural Network we used the Multilayer Perceptron algorithm, as described in
77: \cite{MLPfit}. As explained before in Section \ref{sub:Results-of-the}, the first 1400000
78: events in the ``loose-tight'' sample were used as background
1.3 ! uid12904 79: 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.
1.1 uid12904 80: 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
81: for the measurement.
82: When doing the measurement later on (Section \ref{sub:xsect}) we pick the tau with the highest $NN(\tau)$
83: in the signal sample as the tau cadidate at same time that taus in the loose-tight sample are picked at
84: random since all of them are regarded as fake taus by being below the cut $NN(\tau)$ = 0.7. By doing this
85: we expect to avoid any bias when selecting real taus for the measurement.
86: Figures \ref{fig:nnout_type2_training} and \ref{fig:nnout_type3_training} show the
87: effect of each of the chosen the topological event NN input variables on the final output.
88:
89: Figures \ref{fig:nnout_type2} and \ref{fig:nnout_type3} show the NN output as a result of the training
90: described above. It is evident from both pictures that high values of NN correspond to
91: the signal-enriched region.
92:
93:
94:
95: \begin{figure}[h]
1.3 ! uid12904 96: \includegraphics[scale=0.49]{plots/SetI_NNout_SM_type2_tauQCD.eps}
! 97: \caption{Training of topological Neural Network output for Type 1 and 2 $\tau$ channel combined.
! 98: Upper left: relative impact of each of the input variables; upper right: relative weights
! 99: of the synaptic connections of the trained network;
! 100: lower left: convergence curves; lower right: the output distribution of signal and background
! 101: test samples after training.}
1.1 uid12904 102: \label{fig:nnout_type2_training}
103: \end{figure}
104:
1.3 ! uid12904 105: %\newpage
1.1 uid12904 106:
107:
108: \begin{figure}[h]
1.3 ! uid12904 109: \includegraphics[scale=0.49]{plots/SetI_NNout_SM_type3_tauQCD.eps}
1.1 uid12904 110: \caption{Training of topological Neural Network output for type 3 $\tau$ channel.
1.3 ! uid12904 111: Upper left: relative impact of each of the input variables; upper right: relative weights
! 112: of the synaptic connections of the trained network;
! 113: lower left: convergence curves; lower right: the output distribution of signal and background
! 114: test samples after training.}
1.1 uid12904 115: \label{fig:nnout_type3_training}
116: \end{figure}
117:
118: \begin{figure}[h]
119: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeI_II/nnout.eps}
120: \caption{The topological Neural Network output for type 1 and 2 $\tau$ channel}
121: \label{fig:nnout_type2}
122: \end{figure}
123:
124: \newpage
125:
126: \begin{figure}[t]
127: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeIII/nnout.eps}
128: \caption{The topological Neural Network output for type 3 $\tau$ channel}
129: \label{fig:nnout_type3}
130: \end{figure}
131:
132:
133: \subsection{\label{sub:NN-optimization}NN optimization}
134: One difference between this present analysis and the previous p17 is that we performed a NN optimization along with a
135: $\not\!\! E_{T}$ significance optimization. Previously a cut of $>$ 3.0 was applied to $\not\!\! E_{T}$ significance
136: at the preselection stage and then it was included as one of the variables for NN training. This time as we
137: chose to optimize it, since it is still a good variable to provide signal-background discrimination (Figure \ref{fig:metl_note}).
138: It is important to stress out that after the optimization we performed the
139: analysis with the optimized $\not\!\! E_{T}$ significance cut
140: applied when doing both $\tau$ and b ID (Section \ref{sub:Results-of-the}), therefore
141: after the preselection where no $\not\!\! E_{T}$ significance cut was applied.
142: We then went back anp reprocessed (preselected) all MC samples with the optimized cut. Both results,
143: with $\not\!\! E_{T}$ significance applied during and after preselectio were identical.
144: We then chose to present this analyis with this cut applied at the preselection
145: level in order to have a consistent cut flow throughout the analysis(Section \ref{sub:Preselection}).
146:
147:
148: \begin{figure}[h]
149: \includegraphics[scale=0.5]{plots/metl_allEW.eps}
150: \caption{$\not\!\! E_{T}$ significance distribution for signal and backgrounds.}
151: \label{fig:metl_note}
152: \end{figure}
153:
154:
155: \newpage
156:
157: Below we describe how we split this part of the analysis into two parts:
158:
159: \begin{enumerate}
1.3 ! uid12904 160: \item {\bf Set optimization:} We applied an ``reasonable'' cut on $\not\!\! E_{T}$ significance of $\geq$ 4.0 and
! 161: varied the set of varibles going into NN training.
! 162: \item {\bf $\not\!\! E_{T}$ significance optimization:} After chosing the best set based on the lowest RMS of the
! 163: figure of merith used (see Eq. \ref{merit}), we then optimized the $\not\!\! E_{T}$ significance cut.
1.1 uid12904 164: \end{enumerate}
165:
1.3 ! uid12904 166: For this part of the analysis we present the sets of variables that were taken into account to perform the NN traning:
1.1 uid12904 167: \begin{itemize}
1.3 ! uid12904 168: \item \textit{\textbf{Set 1}} : {$H_{T}$}, aplan (aplanarity), Mjjtau ($M_{jj\tau}$)
! 169: \item \textit{\textbf{Set 2}} : {$H_{T}$}, aplan, cent (centrality)
! 170: \item \textit{\textbf{Set 3}} : {$H_{T}$}, aplan, spher (spherecity)
! 171: \item \textit{\textbf{Set 4}} : {$H_{T}$}, cent, spher
! 172: \item \textit{\textbf{Set 5}} : aplan, cent, spher
! 173: \item \textit{\textbf{Set 6}} : {$H_{T}$}, aplan, Mjjtau, spher
! 174: \item \textit{\textbf{Set 7}} : {$H_{T}$}, aplan, Mjjtau, cent
! 175: \item \textit{\textbf{Set 8}} : {$H_{T}$}, aplan, Mjjtau, costhetastar ($cos(\theta^{*})$)
! 176: \item \textit{\textbf{Set 9}} : {$H_{T}$}, aplan, Mjjtau, cent, spher
! 177: \item \textit{\textbf{Set 10}} : {$H_{T}$}, aplan, Mjjtau, cent, costhetastar
! 178: \item \textit{\textbf{Set 11}} : {$H_{T}$}, aplan, Mjjtau, spher, costhetastar
! 179: \item \textit{\textbf{Set 12}} : METsig ($\not\!\! E_{T}$ significance), {$H_{T}$}, aplan, Mjjtau
! 180: \item \textit{\textbf{Set 13}} : METsig, {$H_{T}$}, aplan, cent
! 181: \item \textit{\textbf{Set 14}} : METsig, {$H_{T}$}, aplan, spher
! 182: \item \textit{\textbf{Set 15}} : METsig, {$H_{T}$}, cent, spher
! 183: \item \textit{\textbf{Set 16}} : METsig, {$H_{T}$}, aplan
! 184: \item \textit{\textbf{Set 17}} : METsig, {$H_{T}$}, Mjjtau
! 185: \item \textit{\textbf{Set 18}} : METsig, aplan, Mjjtau
! 186: \item \textit{\textbf{Set 19}} : METsig, {$H_{T}$}, cent
! 187: \item \textit{\textbf{Set 20}} : METsig, {$H_{T}$}, aplan, Mjjtau, cent
! 188: \item \textit{\textbf{Set 21}} : METsig, {$H_{T}$}, aplan, cent, spher
! 189: \item \textit{\textbf{Set 22}} : METsig, {$H_{T}$}, aplan, Mjjtau, spher
! 190: \item \textit{\textbf{Set 23}} : METsig, {$H_{T}$}, aplan, Mjjtau, costhetastar
! 191: \item \textit{\textbf{Set 24}} : METsig, Mjjtau, cent, spher, costhetastar
! 192: \item \textit{\textbf{Set 25}} : METsig, {$H_{T}$}, cent, spher, costhetastar
! 193: \item \textit{\textbf{Set 26}} : METsig, aplan, cent, spher, costhetastar
! 194: \item \textit{\textbf{Set 27}} : METsig, {$H_{T}$}, aplan, cent, costhetastar
! 195: \item \textit{\textbf{Set 28}} : {$H_{T}$}, aplan, topmassl
! 196: \item \textit{\textbf{Set 29}} : {$H_{T}$}, aplan, Mjjtau, topmassl
! 197: \item \textit{\textbf{Set 30}} : {$H_{T}$}, aplan, Mjjtau, cent, topmassl
! 198: \item \textit{\textbf{Set 31}} : {$H_{T}$}, aplan, Mjjtau, costhetastar, topmassl
! 199: \item \textit{\textbf{Set 32}} : METsig, {$H_{T}$}, topmassl, aplan, Mjjtau
! 200: \item \textit{\textbf{Set 33}} : METsig, spher, costhetastar, aplan, cent
! 201: % \item \textit{\textbf{Set XXXIV}} : metl, spher, Mjjtau, topmassl, ktminp
1.1 uid12904 202: \end{itemize}
203:
1.3 ! uid12904 204: P17 tried only three different sets among hundreds of possible combinations. We believe that the
! 205: 33 sets tested above suffice in giving an optimal result.
1.1 uid12904 206: The criteria used for making a decision on which variable should be used follow:
207: \begin{itemize}
1.3 ! uid12904 208: \item No more than 5 variables to keep NN simple and stable. More require larger training samples.
! 209: \item We want to use METsig variable, since it's the one providing best discrimination.
! 210: \item We do not want to use highly correlated variables in same NN. Such as $H_{T}$ and jet $p_{T}$'.
1.1 uid12904 211: % \item We can not use tau-based variables.
212: \item We want to use variables with high discriminating power.
213: \end{itemize}
214:
1.3 ! uid12904 215: In order to make the decision about which of these 33 choices is the optimal we created an ensemble of
1.1 uid12904 216: 20000 pseudo-datasets each containing events randomly (according to a Poisson distribution) picked
217: from QCD, EW and $\ttbar$ templates. Each of these datasets was treated like real data, meaning applying all
218: the cuts and doing the shape fit of event topological NN. QCD templates for fit were made from the same
219: ``loose-tight $\tau$ sample'' from which the QCD component of the ``data'' was drawn.
1.3 ! uid12904 220: We used the folloing quantity as the figure of merit:
1.1 uid12904 221:
222: \begin{equation}
223: f = \displaystyle \frac{(N_{fit} - N_{true})}{N_{true}}
224: \label{merit}
225: \end{equation}
226:
227:
228: \noindent where $N_{fit}$ is the number of $t\bar{t}$ pairs given by
229: the fit and $N_{true}$ is the number of $t\bar{t}$ pairs from the Poisson distribution.
230: In both Set and $\not\!\! E_{T}$ significance optimization, the lowest RMS was used
1.3 ! uid12904 231: to characterize which configuration is the best in each case.
1.1 uid12904 232:
233:
1.3 ! uid12904 234: The plots showing results concerning the set optimization are found in Appendix \ref{app:set_opt} and are summarized
! 235: in Table \ref{setopt_table} below, where each RMS and mean are shown. The parenthesis after each set ID show the number of
1.1 uid12904 236: hidden nodes in NN training.
237:
238: \begin{table}[htbp]
239: \begin{tabular}{|c|r|r|r|} \hline
240: Set of variables & \multicolumn{1}{c|}{RMS} & \multicolumn{1}{c|}{mean} \\ \hline
241:
242: \hline
243:
244:
245: Set1(6) & \multicolumn{1}{c|}{0.1642} & \multicolumn{1}{c|}{0.0265}\\ \hline
246:
247: Set2(6) & \multicolumn{1}{c|}{0.1840} & \multicolumn{1}{c|}{0.0054}\\ \hline
248:
249: Set3(6) & \multicolumn{1}{c|}{0.1923} & \multicolumn{1}{c|}{0.0060}\\ \hline
250:
251: Set4(6) & \multicolumn{1}{c|}{0.1978} & \multicolumn{1}{c|}{0.0175}\\ \hline
252:
253: Set5(6) & \multicolumn{1}{c|}{0.2385} & \multicolumn{1}{c|}{0.0022}\\ \hline
254:
255: Set6(8) & \multicolumn{1}{c|}{0.1687} & \multicolumn{1}{c|}{0.0115}\\ \hline
256:
257: Set7(8) & \multicolumn{1}{c|}{0.1667} & \multicolumn{1}{c|}{0.0134}\\ \hline
258:
259: Set8(10) & \multicolumn{1}{c|}{0.1668} & \multicolumn{1}{c|}{0.0162}\\ \hline
260:
261: Set9(10) & \multicolumn{1}{c|}{0.1721} & \multicolumn{1}{c|}{0.0102}\\ \hline
262:
263: Set10(10) & \multicolumn{1}{c|}{0.1722} & \multicolumn{1}{c|}{0.0210}\\ \hline
264:
265: Se11(10) & \multicolumn{1}{c|}{0.1716} & \multicolumn{1}{c|}{0.0180}\\ \hline
266:
267: Set12(8) & \multicolumn{1}{c|}{0.1662} & \multicolumn{1}{c|}{0.0039}\\ \hline
268:
269: Set13(8) & \multicolumn{1}{c|}{0.1819} & \multicolumn{1}{c|}{0.0018}\\ \hline
270:
271: Set14(8) & \multicolumn{1}{c|}{0.1879} & \multicolumn{1}{c|}{0.0019}\\ \hline
272:
273: Set15(8) & \multicolumn{1}{c|}{0.1884} & \multicolumn{1}{c|}{-0.0004}\\ \hline
274:
275: Set16(6) & \multicolumn{1}{c|}{0.1912} & \multicolumn{1}{c|}{0.0034}\\ \hline
276:
277: Set17(6) & \multicolumn{1}{c|}{0.1768} & \multicolumn{1}{c|}{0.0074}\\ \hline
278:
279: Set18(6) & \multicolumn{1}{c|}{0.2216} & \multicolumn{1}{c|}{-0.0030}\\ \hline
280:
281: Set19(6) & \multicolumn{1}{c|}{0.1921} & \multicolumn{1}{c|}{0.0015}\\ \hline
282:
283: Set20(10) & \multicolumn{1}{c|}{0.1620} & \multicolumn{1}{c|}{0.0262}\\ \hline
284:
285: Set21(10) & \multicolumn{1}{c|}{0.1753} & \multicolumn{1}{c|}{0.0010}\\ \hline
286:
287: Set22(10) & \multicolumn{1}{c|}{0.1646} & \multicolumn{1}{c|}{0.0086}\\ \hline
288:
289: Set23(10) & \multicolumn{1}{c|}{0.1683} & \multicolumn{1}{c|}{0.0132}\\ \hline
290:
291: Set24(10) & \multicolumn{1}{c|}{0.2053} & \multicolumn{1}{c|}{0.0122}\\ \hline
292:
293: Set25(10) & \multicolumn{1}{c|}{0.1906} & \multicolumn{1}{c|}{0.0038}\\ \hline
294:
295: Set26(10) & \multicolumn{1}{c|}{0.2130} & \multicolumn{1}{c|}{0.0028}\\ \hline
296:
297: Set27(10) & \multicolumn{1}{c|}{0.1859} & \multicolumn{1}{c|}{0.0004}\\ \hline
298:
299: Set28(6) & \multicolumn{1}{c|}{0.1910} & \multicolumn{1}{c|}{-0.0022}\\ \hline
300:
301: Set29(8) & \multicolumn{1}{c|}{0.1587} & \multicolumn{1}{c|}{0.0214}\\ \hline
302:
303: Set30(10) & \multicolumn{1}{c|}{0.1546} & \multicolumn{1}{c|}{0.0148}\\ \hline
304:
305: Set31(10) & \multicolumn{1}{c|}{0.1543} & \multicolumn{1}{c|}{0.0203}\\ \hline
306:
307: Set32(10) & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
308:
309: Set33(10) & \multicolumn{1}{c|}{0.2201} & \multicolumn{1}{c|}{0.0081}\\ \hline
310:
311: %Set34(10) & \multicolumn{1}{c|}{0.1955} & \multicolumn{1}{c|}{0.0184}\\ \hline
312: \end{tabular}
313: \caption{Results for set optimization part whit $\not\!\! E_{T}$ significance $>$ 4.0 applied to all sets.
314: The number in parenthesis refers to number of hidden nodes in each case.}
315: \label{setopt_table}
316: \end{table}
317:
1.3 ! uid12904 318: From Table \ref{setopt_table} we see that Set 32 has the lowest RMS, thus we chose it
1.1 uid12904 319: as the set to be used in $\not\!\! E_{T}$ significance optimization part, whose results are
320: shown in Appendix \ref{app:metl_opt} and then summarized in Table \ref{metlopt_table} below
321:
322: \begin{table}[htbp]
1.3 ! uid12904 323: \begin{tabular}{|c|r|r|r|r|} \hline
! 324: Set 32 & Number of hidden nodes &$\not\!\! E_{T}$ significance cut & RMS & \multicolumn{1}{c|}{mean} \\ \hline
1.1 uid12904 325:
326: \hline
327:
328:
329: %Set6(10) & \multicolumn{1}{c|}{1.0} & \multicolumn{1}{c|}{0.2611} \\ \hline
330:
331: %Set6(10) & \multicolumn{1}{c|}{1.5} & \multicolumn{1}{c|}{0.2320} \\ \hline
332:
333: %Set6(10) & \multicolumn{1}{c|}{2.0} & \multicolumn{1}{c|}{0.2102} \\ \hline
334:
335: %Set6(10) & \multicolumn{1}{c|}{2.5} & \multicolumn{1}{c|}{0.2021} \\ \hline
336:
1.3 ! uid12904 337: 1 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{3.0} & \multicolumn{1}{c|}{0.1507} & \multicolumn{1}{c|}{0.0157}\\ \hline
1.1 uid12904 338:
1.3 ! uid12904 339: 2 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{3.5} & \multicolumn{1}{c|}{0.1559} & \multicolumn{1}{c|}{0.0189}\\ \hline
1.1 uid12904 340:
1.3 ! uid12904 341: 3 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{4.0} & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
1.1 uid12904 342:
1.3 ! uid12904 343: 4 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{4.5} & \multicolumn{1}{c|}{0.1511} & \multicolumn{1}{c|}{0.0153}\\ \hline
1.1 uid12904 344:
1.3 ! uid12904 345: 5 & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{5.0} & \multicolumn{1}{c|}{0.1552} & \multicolumn{1}{c|}{0.0205}\\ \hline
1.1 uid12904 346:
347: %Set6(10) & \multicolumn{1}{c|}{5.5} & \multicolumn{1}{c|}{0.4008} \\ \hline
348: \end{tabular}
349: \caption{Results for $\not\!\! E_{T}$ significance optimization part when varying the $\not\!\! E_{T}$ significance cut
350: The number in parenthesis refers to number of hidden nodes in each case.}
351: \label{metlopt_table}
352: \end{table}
353:
354:
355:
356: Combined results from Tables \ref{setopt_table} and \ref{metlopt_table} show that the best configuration found
1.3 ! uid12904 357: was Set 32 with $\not\!\! E_{T}$ significance $\geq$ 4.0. Therefore, this was the
! 358: configuration used to perform the cross section measurement.Figure \ref{fig:METsig_RMS} shows the variation of the RMS as function
1.1 uid12904 359: of the $\not\!\! E_{T}$ significance we applied.
360:
1.3 ! uid12904 361: \begin{figure}[h]
! 362: \includegraphics[scale=0.35]{plots/METsig-RMS.eps}
1.1 uid12904 363: \caption{Plot of RMS as a function the $\not\!\! E_{T}$ significance applied}
364: \label{fig:METsig_RMS}
365: \end{figure}
366:
367:
1.3 ! uid12904 368: %\clearpage
1.1 uid12904 369:
370:
371: In order to check the validity of our emsemble tests procedure, it is instructive to plot both the
372: distribution of the predicted number of $t\bar{t}$ and what is called ``pull'', defined in Equation
373: \ref{pull} below:
374:
375: \begin{equation}
376: p = \displaystyle \frac{(N_{fit}-N_{true})}{\sigma_{fit}}
377: \label{pull}
378: \end{equation}
379:
380: \noindent where $\sigma_{fit}$ is the error on the number of $t\bar{t}$ pairs given by the fit.
381:
382: Figures \ref{fig:gaus_ttbar} and \ref{fig:pull} show both beforementioned distributions.
383:
384: From Figure \ref{fig:gaus_ttbar} we see a good agreement between the number of $t\bar{t}$ pairs
385: initially set in the ensemble and the measured value. And Figure \ref{fig:pull} shows a nice gaussian
386: curve, that indicates a good behaviour of the fit uncertainties in the ensembles.
387:
388: \begin{figure}[t]
1.3 ! uid12904 389: \includegraphics[scale=0.40]{plots/gaus_ttbar.eps}
1.1 uid12904 390: \caption{Distribution of the output ``measurement'' for an ensemble with 116.9 $\ttbar$ events.}
391: \label{fig:gaus_ttbar}
392: \end{figure}
393:
1.3 ! uid12904 394: \begin{figure}[b]
! 395: \includegraphics[scale=0.40]{plots/pull1-40.eps}
1.1 uid12904 396: \caption{The ensemble test's pull.}
397: \label{fig:pull}
398: \end{figure}
399:
400:
401: %\newpage
402:
403:
404:
405:
406:
407: \clearpage
408:
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