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Presentations / Zurich_group / 30_06_2014 / MMatrix.tex~
@mchrzasz mchrzasz on 13 Aug 2014 28 KB update
  1. \documentclass[xcolor=svgnames]{beamer}
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  11. \usepackage{hepnicenames}
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  21. \usetheme{Sybila}
  22.  
  23. \title[Unfolding for counting experiments]{Unfolding for counting experiments}
  24. \author{Marcin Chrz\k{a}szcz$^{1,2}$, Nicola Serra$^{1}$}
  25. \institute{$^1$~University of Zurich,\\ $^2$~Institute of Nuclear Physics, Krakow}
  26. \date{\today}
  27.  
  28. \begin{document}
  29. % --------------------------- SLIDE --------------------------------------------
  30. \frame[plain]{\titlepage}
  31. \author{Marcin Chrz\k{a}szcz}
  32. % ------------------------------------------------------------------------------
  33. % --------------------------- SLIDE --------------------------------------------
  34.  
  35. \begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
  36.  
  37. \center \includegraphics[width=0.8\paperwidth]{diagram.png}\\
  38.  
  39. \end{frame}
  40. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  41. \begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
  42.  
  43. \center \includegraphics[width=0.8\paperwidth]{matrix.png}\\
  44.  
  45. \end{frame}
  46.  
  47.  
  48. \section{Introduction}
  49.  
  50. \begin{frame}\frametitle{Reminder}
  51. For now:
  52. \begin{itemize}
  53. \item We have proven that there has to exists unfolding matrix.
  54. \item Shown how to construct transformation matrix: $Gen \to Reco$.
  55. \item Inverting it we can have transformation matrix of $Reco \to Gen$.
  56. \item For details: \href{https://indico.cern.ch/event/316905/session/1/contribution/18/material/slides/0.pdf}{LINK}
  57.  
  58. \end{itemize}
  59.  
  60. What is missing?
  61. \begin{columns}
  62. \column{1in}
  63. \begin{enumerate}
  64. \item ERROR!
  65. \end{enumerate}
  66. \column{4in}
  67. \includegraphics[width=0.8\textwidth]{err.jpg}\\
  68. \end{columns}
  69.  
  70. \end{frame}
  71.  
  72.  
  73.  
  74. \begin{frame}\frametitle{How to?}
  75. \begin{itemize}
  76. \item So lets say that transformation matrix:$Gen \to Reco$ is $\epsilon_{i,j}$.
  77. \item Of coz it's easy to write the covariance matrix(error matrix):
  78. \end{itemize}
  79. \begin{equation}
  80. cov(\epsilon_{\alpha, \beta},\epsilon_{a,b})
  81. \end{equation}
  82. \begin{itemize}
  83. \item Then we can calculate the matrix: $\epsilon_{i,j}^{-1}$(assuming it exists).
  84. \item and the 1M dollar question is: $cov(\epsilon_{\alpha, \beta}^{-1},\epsilon_{a,b}^{-1}) =?$
  85. \end{itemize}
  86.  
  87. \end{frame}
  88.  
  89. \begin{frame}\frametitle{Answer to 1M dolar quesion}
  90. \begin{itemize}
  91. \item Solution comes from $\tau$ physics :)
  92. \end{itemize}
  93. \begin{equation}
  94. cov(\epsilon_{\alpha, \beta}^{-1},\epsilon_{a,b}^{-1}) = \epsilon^{-1}_{\alpha,i} \epsilon^{-1}_{j,\beta} \epsilon^{-1}_{a,k} \epsilon^{-1}_{l,b} cov( \epsilon_{ij} \epsilon_{kl})
  95. \end{equation}
  96. \begin{itemize}
  97. \item It's way to late to latex the prove. For prove see: \href{http://arxiv.org/abs/hep-ex/9909031}{arXiv:hep-ex/9909031}
  98. \item Thanks to orthonormal basis lifes gets simpler:
  99. \end{itemize}
  100. \begin{equation}
  101. cov(\epsilon_{ij},\epsilon_{kl} ) = \sigma_{\epsilon,ij} \delta_{ik} \delta_{jl} ( no correlations)
  102. \end{equation}
  103.  
  104. \end{frame}
  105.  
  106.  
  107. \begin{frame}\frametitle{Total error}
  108. \begin{itemize}
  109. \item So let's say: $B=\epsilon^{-1} f$
  110. \item Then:
  111. \end{itemize}
  112. \begin{equation}
  113. cov(B_i,B_j) = f_a f_b cov(\epsilon_{i \alpha }^{-1} , \epsilon_{j \beta}^{-1} )+ \epsilon^{-1}_{ ik} \epsilon^{-1}_{jl} cov(f_k,f_l)
  114. \end{equation}
  115. \begin{itemize}
  116. \item Looks easy just need to implement!
  117. \end{itemize}
  118.  
  119. \end{frame}
  120.  
  121.  
  122. \begin{frame}\frametitle{Matrix, $0.1-0.98~GeV$}
  123. \tiny{
  124. $ A_{reco\rightarrow gen}=\begin{pmatrix}
  125. 0.9495 0.008518 0.01522 -0.007362 0.01496 -0.04544 -0.02468 0.002078
  126. 0.0002651 0.8261 0.00978 -0.001042 0.004382 0.03191 0.01247 0.01886
  127. -0.0003391 0.009773 1.03 0.008433 -0.002028 0.003 0.02086 0.001549
  128. -0.00403 -0.001336 0.01047 0.9215 0.006898 -0.0006023 0.003041 -0.004465
  129. 0.006557 0.005531 -0.002423 0.00691 1.171 -0.01082 -0.0158 -0.003231
  130. -0.01931 0.03973 0.00342 -0.0006455 -0.01081 0.9452 0.02232 -0.005304
  131. .008421 0.0124 0.02071 0.002415 -0.01264 0.01785 1.06 0.001084
  132. 0.000347 0.01886 0.001555 -0.003568 -0.002591 -0.004226 0.001093 0.8204
  133. \end{pmatrix}$
  134. }
  135.  
  136.  
  137.  
  138.  
  139.  
  140. \end{frame}
  141.  
  142. \begin{frame}\frametitle{Constructing Matrix unfolding}
  143. \begin{itemize}
  144. \item We got first column of the unfolding matrix.
  145. \end{itemize}
  146. \small{
  147. $ \begin{pmatrix}
  148. 1.06 & \cdots & a_{1,8} \\
  149. 0.01157 & \cdots & a_{2,8} \\
  150. -0.003547 & \ddots & \vdots \\
  151. 0.0007841 & \ddots & \vdots \\
  152. 0.001126 & \ddots & \vdots \\
  153. 0.001766 & \ddots & \vdots \\
  154. 0.001664 & \ddots & \vdots \\
  155. -0.001937 & \cdots & a_{8,8}
  156. \end{pmatrix}$
  157.  
  158.  
  159. }
  160. \begin{itemize}
  161. \item How about the others?
  162. \item We can reweight accordingly to $f_x$.
  163. \end{itemize}
  164.  
  165. \end{frame}
  166.  
  167. \begin{frame}\frametitle{Constructing Matrix unfolding}
  168. \begin{itemize}
  169. \item To get $S_3$ each event $i^{th}$ has has weight $f_{S_3}(\cos \theta_{k_i},\cos \theta_{l_i},\phi_i) $
  170. \item One can calculate on MC the reweighed moments in PHPS:
  171. \end{itemize}
  172. \begin{equation}
  173. \int PDF*f_{S_3}=\dfrac{32}{225}
  174. \end{equation}
  175. \begin{itemize}
  176. \item Our base vector now is:$v^{T}_{gen}=(0 ,\frac{32}{225},0,0,0,0,0,0)$
  177. \item So lets see what do we get as reconstructed vector(after multiplying by $\frac{225}{32}$.
  178. \small{$v^{T}_{rec}=( 0.042, 1.105,-0.005,0.003,-0.0023,-0.005,-0.005,-0.006)$ }
  179. \end{itemize}
  180.  
  181. \end{frame}
  182.  
  183. \begin{frame}\frametitle{Constructing Matrix unfolding}
  184. \begin{itemize}
  185. \item Now the matrix looks like:
  186. \end{itemize}
  187. \small{
  188. $ \begin{pmatrix}
  189. 1.06 & 0.042 & \cdots & a_{1,8} \\
  190. 0.01157 & 1.105 & \cdots & a_{2,8} \\
  191. -0.003547 & -0.005 & \ddots & \vdots \\
  192. 0.0007841 &-0.005 & \ddots & \vdots \\
  193. 0.001126 & 0.003 &\ddots & \vdots \\
  194. 0.001766 & -0.0023 &\ddots & \vdots \\
  195. 0.001664 & -0.005 &\ddots & \vdots \\
  196. -0.001937 & -0.006 &\cdots & a_{8,8}
  197. \end{pmatrix}$
  198.  
  199.  
  200. }
  201. \begin{itemize}
  202. \item The others go in the same way.
  203. \item Repenting this exercise from $1^{st}$ year algebra we can get the full matrix
  204. \end{itemize}
  205.  
  206.  
  207. \end{frame}
  208. \begin{frame}\frametitle{Constructing Matrix unfolding}
  209. \begin{itemize}
  210. \item The full transformation matrix from generator space to reconstructed space:
  211. \end{itemize}
  212. \tiny{
  213. $ A_{gen\rightarrow reco}=\begin{pmatrix}
  214. 1.06 & 0.0423 & -0.0081 & 0.0022 & 0.0049 & 0.0037 & 0.0028 & -0.0065 \\
  215.  
  216. 0.0115 & 1.105 & -0.0050 & 0.0027 & -0.0018 & -0.0040 &-0.0054 & -0.0065 \\
  217. -0.0035 & -0.0050 & 0.981 & 0.0005 & -0.0025 & 0.0002 & -0.0037 & -0.0048\\
  218. 0.00078 & 0.0034 & 0.0006 & 1.002 & -0.0032 & -0.0040 & 0.0003 & 0.0018\\
  219. 0.001126 & -0.0023 & -0.0032 & -0.0032 & 1.055 & 0.001& -0.004 & 0.0023\\
  220. 0.00176 & -0.0050 & 0.00036 & -0.0040 & 0.0011 & 0.96 & -0.0057 & 0.0009 \\
  221. 0.0016 & -0.005 & -0.003 & 0.00029& -0.003 &-0.004 & 0.9543 & 0.0000\\
  222. -0.0019 & -0.0065 & -0.004 & 0.001 & 0.0018 & 0.0007 & 0.000 & 1.098 \\
  223.  
  224. \end{pmatrix}$
  225.  
  226.  
  227. }
  228. \begin{itemize}
  229. \item Inverting the matrix is simple, and doable
  230. \end{itemize}
  231. \tiny{
  232. $ A_{reco\rightarrow gen}=\begin{pmatrix}
  233. 0.9434 & -0.036 & 0.007& -0.0020 & -0.0044& -0.0038 & -0.0030 & 0.0054\\
  234. -0.009 & 0.90 & 0.0045 & -0.0024& 0.0016 & 0.003873 & 0.00527& 0.005 \\
  235. 0.003 & 0.00454& 1.019 & -0.00058 & 0.0025& -0.000291 & 0.004 & 0.004 \\
  236. -0.00071 & -0.0030 & -0.0007 & 0.9977 & 0.0030 & 0.004206 &-0.0003 & -0.0017 \\
  237. -0.001 & 0.0020 & 0.0031& 0.0030 & 0.9483 & -0.0010 & 0.004626 & -0.0019 \\
  238. -0.001 & 0.004 & -0.0003 & 0.0042 & -0.001087 & 1.037 & 0.0063 & -0.0009\\
  239. -0.0017 & 0.0053 & 0.0042 & -0.0002 & 0.00370& 0.0050 & 1.048 & 0.0000 \\
  240. 0.0016& 0.0053& 0.00452 & -0.001 & -0.001582 & -0.0007213 &0.000 & 0.9105 \\
  241.  
  242. \end{pmatrix}$
  243. }
  244.  
  245. \end{frame}
  246.  
  247. \begin{frame}\frametitle{Sensitivity to unknowns}
  248. \begin{itemize}
  249. \item We are unfolding based on MC.
  250. \item There are MC/Data differences, which can have impact on the unfolding.
  251. \end{itemize}
  252. Let's put small modification:
  253. \begin{equation}
  254. w_j \to \overline{w_j}= \dfrac{1}{eff(\cos \theta_{l~j}, \cos \theta_{k~j}, \phi_{j})} \times corr(\cos \theta_{l~j}, \cos \theta_{k~j}, \phi_{j})
  255. \end{equation}
  256. Unfortunately God didn't allowed me sneak peak into his cards so I don't know $corr(\cos \theta_{l}, \cos \theta_{k}, \phi)$, but let's try out some functions and see what happens :)
  257.  
  258.  
  259. \end{frame}
  260.  
  261. \begin{frame}\frametitle{Corr1 functions}
  262. \begin{columns}
  263. \column{2in}
  264. \includegraphics[width=\linewidth]{corr/Corr1_cosk.png}\\
  265. \includegraphics[width=\linewidth]{corr/Corr1_cosl.png}\\
  266. \column{2.5in}
  267. $
  268. corr1(\cos_l, \cos_k,\phi)= 1+ 0.032 \cos_l - 0.032 \cos_k + 0.01 \phi
  269. $
  270. \includegraphics[width=0.8\linewidth]{corr/Corr1_phi.png}\\
  271.  
  272. \end{columns}
  273.  
  274. \end{frame}
  275. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  276. \begin{frame}\frametitle{Corr2 functions}
  277. \begin{columns}
  278. \column{2in}
  279. \includegraphics[width=0.85\linewidth]{corr/corr21.png}\\
  280. \includegraphics[width=0.85\linewidth]{corr/corr22.png}\\
  281. \column{2.5in}
  282. $
  283. corr2(\cos_l, \cos_k,\phi)= -0.02 \cos_l^2 + 0.02 \cos_k^2 -
  284. 0.015 \phi^2+ 1
  285. $
  286. \includegraphics[width=0.75\linewidth]{corr/corr23.png}\\
  287.  
  288. \end{columns}
  289.  
  290. \end{frame}
  291.  
  292. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  293. \begin{frame}\frametitle{Corr3 functions}
  294. \begin{columns}
  295. \column{2in}
  296. \includegraphics[width=0.85\linewidth]{corr/corr31.png}\\
  297. \includegraphics[width=0.85\linewidth]{corr/corr32.png}\\
  298. \column{2.5in}
  299. $
  300. corr3(\cos_l, \cos_k,\phi)= 0.02 \cos_l \cos_k + 0.01 \cos_k \phi - 0.01 \phi \cos_l + 1
  301. $
  302. \includegraphics[width=0.75\linewidth]{corr/corr33.png}\\
  303.  
  304. \end{columns}
  305.  
  306. \end{frame}
  307. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  308. \begin{frame}\frametitle{Corr4 functions}
  309. \begin{columns}
  310. \column{2in}
  311. \includegraphics[width=0.85\linewidth]{corr/corr41.png}\\
  312. \includegraphics[width=0.85\linewidth]{corr/corr42.png}\\
  313. \column{2.5in}
  314. $
  315. corr3(\cos_l, \cos_k,\phi)= 0.01 \cos_k \cos_l \phi + 1
  316. $
  317. \includegraphics[width=0.75\linewidth]{corr/corr43.png}\\
  318.  
  319. \end{columns}
  320.  
  321. \end{frame}
  322.  
  323.  
  324.  
  325.  
  326. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  327.  
  328.  
  329. \begin{frame}\frametitle{Corr1- MM}
  330.  
  331.  
  332. \begin{tiny}
  333.  
  334. \begin{center}
  335. \begin{tabular}{ l l l l l l l l l }
  336. \hline
  337. \multicolumn{8}{c}{Mean of the pull} \\ \hline
  338. $q^2$ & $S_3$ & $S_4$ & $S_5$ & $S_{6s}$ & $S_{6c}$ & $S_7$ & $S_8$ \\ \hline \hline
  339.  
  340. 0 & \scalebox{0.5}{$0.0085 \pm 0.026(0.3)$} & \scalebox{0.5}{$0.13 \pm 0.027(4.6)$} & \scalebox{0.5}{$-0.025 \pm 0.027(-0.92)$} & \scalebox{0.5}{$0.46 \pm 0.027(17)$} & \scalebox{0.5}{$-0.13 \pm 0.028(-4.7)$} & \scalebox{0.5}{$0.029 \pm 0.028(1)$} & \scalebox{0.5}{$-0.66 \pm 0.027(-25)$} \\ \hline
  341. 1 & \scalebox{0.5}{$0.0094 \pm 0.028(0.33)$} & \scalebox{0.5}{$0.078 \pm 0.028(2.8)$} & \scalebox{0.5}{$-0.02 \pm 0.028(-0.73)$} & \scalebox{0.5}{$0.24 \pm 0.028(8.5)$} & \scalebox{0.5}{$-0.075 \pm 0.028(-2.7)$} & \scalebox{0.5}{$-0.016 \pm 0.026(-0.63)$} & \scalebox{0.5}{$-0.39 \pm 0.028(-14)$} \\ \hline
  342. 2 & \scalebox{0.5}{$-0.02 \pm 0.027(-0.72)$} & \scalebox{0.5}{$0.027 \pm 0.026(1)$} & \scalebox{0.5}{$-0.024 \pm 0.027(-0.91)$} & \scalebox{0.5}{$0.18 \pm 0.027(6.8)$} & \scalebox{0.5}{$-0.024 \pm 0.027(-0.89)$} & \scalebox{0.5}{$-0.067 \pm 0.027(-2.5)$} & \scalebox{0.5}{$-0.39 \pm 0.028(-14)$} \\ \hline
  343. 3 & \scalebox{0.5}{$0.013 \pm 0.028(0.46)$} & \scalebox{0.5}{$-0.0089 \pm 0.027(-0.32)$} & \scalebox{0.5}{$0.055 \pm 0.026(2.1)$} & \scalebox{0.5}{$0.12 \pm 0.027(4.7)$} & \scalebox{0.5}{$0.11 \pm 0.027(3.9)$} & \scalebox{0.5}{$-0.018 \pm 0.028(-0.63)$} & \scalebox{0.5}{$-0.43 \pm 0.027(-16)$} \\ \hline
  344. 4 & \scalebox{0.5}{$-0.0054 \pm 0.029(-0.18)$} & \scalebox{0.5}{$-0.066 \pm 0.027(-2.4)$} & \scalebox{0.5}{$0.037 \pm 0.028(1.3)$} & \scalebox{0.5}{$0.22 \pm 0.027(8.1)$} & \scalebox{0.5}{$0.099 \pm 0.027(3.7)$} & \scalebox{0.5}{$-0.1 \pm 0.026(-3.8)$} & \scalebox{0.5}{$-0.41 \pm 0.026(-15)$} \\ \hline
  345. 5 & \scalebox{0.5}{$0.06 \pm 0.027(2.2)$} & \scalebox{0.5}{$-0.069 \pm 0.026(-2.6)$} & \scalebox{0.5}{$-0.013 \pm 0.028(-0.49)$} & \scalebox{0.5}{$0.21 \pm 0.027(7.8)$} & \scalebox{0.5}{$0.093 \pm 0.027(3.5)$} & \scalebox{0.5}{$-0.083 \pm 0.028(-3)$} & \scalebox{0.5}{$-0.41 \pm 0.028(-15)$} \\ \hline
  346. 6 & \scalebox{0.5}{$0.0064 \pm 0.026(0.25)$} & \scalebox{0.5}{$-0.051 \pm 0.027(-1.9)$} & \scalebox{0.5}{$-0.029 \pm 0.028(-1)$} & \scalebox{0.5}{$0.26 \pm 0.028(9.2)$} & \scalebox{0.5}{$0.14 \pm 0.027(5.1)$} & \scalebox{0.5}{$-0.081 \pm 0.027(-3)$} & \scalebox{0.5}{$-0.45 \pm 0.028(-16)$} \\ \hline
  347. %7 & \scalebox{0.5}{$0.22 \pm 0.027(8)$} & \scalebox{0.5}{$-0.34 \pm 0.028(-12)$} & \scalebox{0.5}{$-0.43 \pm 0.026(-16)$} & \scalebox{0.5}{$2.4 \pm 0.029(82)$} & \scalebox{0.5}{$0.66 \pm 0.027(24)$} & \scalebox{0.5}{$-0.24 \pm 0.028(-8.6)$} & \scalebox{0.5}{$-0.51 \pm 0.027(-19)$} &
  348. 8 & \scalebox{0.5}{$0.023 \pm 0.027(0.85)$} & \scalebox{0.5}{$-0.031 \pm 0.028(-1.1)$} & \scalebox{0.5}{$0.0042 \pm 0.028(0.15)$} & \scalebox{0.5}{$0.21 \pm 0.026(7.8)$} & \scalebox{0.5}{$0.12 \pm 0.028(4.2)$} & \scalebox{0.5}{$-0.13 \pm 0.027(-4.8)$} & \scalebox{0.5}{$-0.48 \pm 0.026(-18)$} \\ \hline
  349. 9 & \scalebox{0.5}{$-0.017 \pm 0.027(-0.63)$} & \scalebox{0.5}{$-0.015 \pm 0.026(-0.56)$} & \scalebox{0.5}{$-0.0052 \pm 0.026(-0.2)$} & \scalebox{0.5}{$0.27 \pm 0.027(10)$} & \scalebox{0.5}{$0.046 \pm 0.026(1.7)$} & \scalebox{0.5}{$-0.12 \pm 0.026(-4.4)$} & \scalebox{0.5}{$-0.5 \pm 0.026(-19)$} \\ \hline
  350. 10 & \scalebox{0.5}{$-0.054 \pm 0.027(-2)$} & \scalebox{0.5}{$-0.056 \pm 0.026(-2.2)$} & \scalebox{0.5}{$0.036 \pm 0.026(1.4)$} & \scalebox{0.5}{$0.16 \pm 0.028(5.7)$} & \scalebox{0.5}{$0.077 \pm 0.028(2.8)$} & \scalebox{0.5}{$-0.034 \pm 0.027(-1.3)$} & \scalebox{0.5}{$-0.39 \pm 0.028(-14)$} \\ \hline
  351. 11 & \scalebox{0.5}{$0.023 \pm 0.027(0.88)$} & \scalebox{0.5}{$0.021 \pm 0.027(0.8)$} & \scalebox{0.5}{$-0.011 \pm 0.027(-0.41)$} & \scalebox{0.5}{$0.14 \pm 0.027(5)$} & \scalebox{0.5}{$0.042 \pm 0.027(1.6)$} & \scalebox{0.5}{$-0.098 \pm 0.028(-3.5)$} & \scalebox{0.5}{$-0.3 \pm 0.027(-11)$} \\ \hline
  352.  
  353.  
  354. \hline
  355.  
  356.  
  357. \end{tabular}
  358.  
  359. \end{center}
  360.  
  361.  
  362. \end{tiny}
  363. \end{frame}
  364.  
  365. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  366.  
  367.  
  368. \begin{frame}\frametitle{Corr2- MM}
  369.  
  370.  
  371. \begin{tiny}
  372.  
  373. \begin{center}
  374. \begin{tabular}{ l l l l l l l l l }
  375. \hline
  376. \multicolumn{8}{c}{Mean of the pull} \\ \hline
  377. $q^2$ & $S_3$ & $S_4$ & $S_5$ & $S_{6s}$ & $S_{6c}$ & $S_7$ & $S_8$ \\ \hline \hline
  378.  
  379. 0 & \scalebox{0.5}{$-0.21 \pm 0.026(-8.1)$} & \scalebox{0.5}{$0.061 \pm 0.026(2.3)$} & \scalebox{0.5}{$-0.016 \pm 0.026(-0.63)$} & \scalebox{0.5}{$0.048 \pm 0.026(1.8)$} & \scalebox{0.5}{$-0.0062 \pm 0.027(-0.23)$} & \scalebox{0.5}{$-0.012 \pm 0.028(-0.44)$} & \scalebox{0.5}{$0.0091 \pm 0.026(0.36)$} \\ \hline
  380. 1 & \scalebox{0.5}{$-0.12 \pm 0.028(-4.5)$} & \scalebox{0.5}{$0.032 \pm 0.027(1.2)$} & \scalebox{0.5}{$-0.0071 \pm 0.026(-0.27)$} & \scalebox{0.5}{$0.02 \pm 0.027(0.75)$} & \scalebox{0.5}{$-0.086 \pm 0.027(-3.2)$} & \scalebox{0.5}{$-0.03 \pm 0.025(-1.2)$} & \scalebox{0.5}{$0.021 \pm 0.027(0.79)$} \\ \hline
  381. 2 & \scalebox{0.5}{$-0.15 \pm 0.027(-5.8)$} & \scalebox{0.5}{$-0.013 \pm 0.026(-0.49)$} & \scalebox{0.5}{$0.011 \pm 0.026(0.44)$} & \scalebox{0.5}{$-0.039 \pm 0.026(-1.5)$} & \scalebox{0.5}{$-0.032 \pm 0.027(-1.2)$} & \scalebox{0.5}{$-0.066 \pm 0.027(-2.5)$} & \scalebox{0.5}{$0.018 \pm 0.028(0.64)$} \\ \hline
  382. 3 & \scalebox{0.5}{$-0.15 \pm 0.027(-5.6)$} & \scalebox{0.5}{$0.025 \pm 0.026(0.96)$} & \scalebox{0.5}{$0.016 \pm 0.027(0.58)$} & \scalebox{0.5}{$-0.062 \pm 0.027(-2.3)$} & \scalebox{0.5}{$0.037 \pm 0.026(1.4)$} & \scalebox{0.5}{$0.026 \pm 0.027(0.96)$} & \scalebox{0.5}{$-0.016 \pm 0.027(-0.6)$} \\ \hline
  383. 4 & \scalebox{0.5}{$-0.12 \pm 0.028(-4.5)$} & \scalebox{0.5}{$-0.024 \pm 0.026(-0.92)$} & \scalebox{0.5}{$0.045 \pm 0.027(1.6)$} & \scalebox{0.5}{$0.0075 \pm 0.026(0.29)$} & \scalebox{0.5}{$0.015 \pm 0.027(0.53)$} & \scalebox{0.5}{$-0.036 \pm 0.026(-1.4)$} & \scalebox{0.5}{$0.037 \pm 0.026(1.4)$} \\ \hline
  384. 5 & \scalebox{0.5}{$-0.095 \pm 0.027(-3.6)$} & \scalebox{0.5}{$-0.032 \pm 0.026(-1.2)$} & \scalebox{0.5}{$0.014 \pm 0.026(0.52)$} & \scalebox{0.5}{$-0.013 \pm 0.026(-0.48)$} & \scalebox{0.5}{$-0.0093 \pm 0.027(-0.35)$} & \scalebox{0.5}{$-0.013 \pm 0.026(-0.51)$} & \scalebox{0.5}{$0.042 \pm 0.027(1.6)$} \\ \hline
  385. 6 & \scalebox{0.5}{$-0.17 \pm 0.025(-6.5)$} & \scalebox{0.5}{$0.008 \pm 0.027(0.3)$} & \scalebox{0.5}{$0.012 \pm 0.027(0.45)$} & \scalebox{0.5}{$0.029 \pm 0.027(1.1)$} & \scalebox{0.5}{$0.0072 \pm 0.027(0.27)$} & \scalebox{0.5}{$-0.0012 \pm 0.026(-0.046)$} & \scalebox{0.5}{$0.011 \pm 0.027(0.42)$} \\ \hline
  386. %7 & \scalebox{0.5}{$0.078 \pm 0.026(3)$} & \scalebox{0.5}{$-0.3 \pm 0.026(-11)$} & \scalebox{0.5}{$-0.34 \pm 0.026(-13)$} & \scalebox{0.5}{$2.1 \pm 0.028(73)$} & \scalebox{0.5}{$0.47 \pm 0.026(18)$} & \scalebox{0.5}{$-0.17 \pm 0.027(-6.1)$} & \scalebox{0.5}{$0.0051 \pm 0.027(0.19)$} \\ \hline
  387. 8 & \scalebox{0.5}{$-0.13 \pm 0.026(-5.1)$} & \scalebox{0.5}{$-0.0077 \pm 0.027(-0.28)$} & \scalebox{0.5}{$0.05 \pm 0.027(1.9)$} & \scalebox{0.5}{$-0.03 \pm 0.026(-1.2)$} & \scalebox{0.5}{$-0.012 \pm 0.028(-0.44)$} & \scalebox{0.5}{$-0.046 \pm 0.026(-1.7)$} & \scalebox{0.5}{$0.031 \pm 0.026(1.2)$} \\ \hline
  388. 9 & \scalebox{0.5}{$-0.15 \pm 0.026(-5.7)$} & \scalebox{0.5}{$-0.0083 \pm 0.026(-0.32)$} & \scalebox{0.5}{$0.03 \pm 0.026(1.2)$} & \scalebox{0.5}{$0.044 \pm 0.027(1.6)$} & \scalebox{0.5}{$-0.07 \pm 0.026(-2.7)$} & \scalebox{0.5}{$-0.022 \pm 0.026(-0.84)$} & \scalebox{0.5}{$-0.045 \pm 0.026(-1.7)$} \\ \hline
  389. 10 & \scalebox{0.5}{$-0.15 \pm 0.025(-5.8)$} & \scalebox{0.5}{$-0.032 \pm 0.025(-1.3)$} & \scalebox{0.5}{$0.059 \pm 0.026(2.2)$} & \scalebox{0.5}{$-0.072 \pm 0.028(-2.6)$} & \scalebox{0.5}{$-0.029 \pm 0.027(-1.1)$} & \scalebox{0.5}{$0.064 \pm 0.027(2.4)$} & \scalebox{0.5}{$0.014 \pm 0.027(0.51)$} \\ \hline
  390. 11 & \scalebox{0.5}{$-0.067 \pm 0.026(-2.6)$} & \scalebox{0.5}{$0.017 \pm 0.026(0.65)$} & \scalebox{0.5}{$-0.015 \pm 0.026(-0.56)$} & \scalebox{0.5}{$-0.051 \pm 0.026(-1.9)$} & \scalebox{0.5}{$-0.0086 \pm 0.026(-0.33)$} & \scalebox{0.5}{$-0.018 \pm 0.028(-0.67)$} & \scalebox{0.5}{$0.017 \pm 0.027(0.62)$} \\ \hline
  391.  
  392.  
  393.  
  394.  
  395.  
  396.  
  397. \hline
  398.  
  399.  
  400. \end{tabular}
  401.  
  402. \end{center}
  403.  
  404.  
  405. \end{tiny}
  406. \end{frame}
  407.  
  408.  
  409. \begin{frame}\frametitle{Corr3- MM}
  410.  
  411.  
  412. \begin{tiny}
  413.  
  414. \begin{center}
  415. \begin{tabular}{ l l l l l l l l l }
  416. \hline
  417. \multicolumn{8}{c}{Mean of the pull} \\ \hline
  418. $q^2$ & $S_3$ & $S_4$ & $S_5$ & $S_{6s}$ & $S_{6c}$ & $S_7$ & $S_8$ \\ \hline \hline
  419.  
  420. 0 & \scalebox{0.5}{$-0.021 \pm 0.026(-0.81)$} & \scalebox{0.5}{$0.041 \pm 0.026(1.5)$} & \scalebox{0.5}{$0.009 \pm 0.027(0.34)$} & \scalebox{0.5}{$0.043 \pm 0.026(1.6)$} & \scalebox{0.5}{$0.13 \pm 0.028(4.8)$} & \scalebox{0.5}{$-0.0072 \pm 0.028(-0.26)$} & \scalebox{0.5}{$0.044 \pm 0.026(1.7)$} \\ \hline
  421. 1 & \scalebox{0.5}{$-0.014 \pm 0.028(-0.51)$} & \scalebox{0.5}{$0.03 \pm 0.027(1.1)$} & \scalebox{0.5}{$0.022 \pm 0.027(0.82)$} & \scalebox{0.5}{$0.015 \pm 0.028(0.53)$} & \scalebox{0.5}{$0.078 \pm 0.028(2.8)$} & \scalebox{0.5}{$-0.037 \pm 0.025(-1.4)$} & \scalebox{0.5}{$0.057 \pm 0.027(2.1)$} \\ \hline
  422. 2 & \scalebox{0.5}{$-0.037 \pm 0.027(-1.4)$} & \scalebox{0.5}{$-0.0013 \pm 0.027(-0.048)$} & \scalebox{0.5}{$-0.015 \pm 0.027(-0.54)$} & \scalebox{0.5}{$-0.052 \pm 0.026(-2)$} & \scalebox{0.5}{$0.1 \pm 0.027(3.8)$} & \scalebox{0.5}{$-0.051 \pm 0.026(-1.9)$} & \scalebox{0.5}{$0.022 \pm 0.028(0.78)$} \\ \hline
  423. 3 & \scalebox{0.5}{$-0.015 \pm 0.027(-0.55)$} & \scalebox{0.5}{$0.036 \pm 0.028(1.3)$} & \scalebox{0.5}{$-0.039 \pm 0.027(-1.4)$} & \scalebox{0.5}{$-0.072 \pm 0.027(-2.7)$} & \scalebox{0.5}{$0.17 \pm 0.027(6.2)$} & \scalebox{0.5}{$-0.0044 \pm 0.028(-0.15)$} & \scalebox{0.5}{$-0.024 \pm 0.027(-0.9)$} \\ \hline
  424. 4 & \scalebox{0.5}{$-0.00047 \pm 0.029(-0.017)$} & \scalebox{0.5}{$-0.012 \pm 0.027(-0.43)$} & \scalebox{0.5}{$0.0099 \pm 0.028(0.35)$} & \scalebox{0.5}{$-0.002 \pm 0.026(-0.076)$} & \scalebox{0.5}{$0.17 \pm 0.028(6.2)$} & \scalebox{0.5}{$-0.062 \pm 0.027(-2.3)$} & \scalebox{0.5}{$0.0086 \pm 0.027(0.32)$} \\ \hline
  425. 5 & \scalebox{0.5}{$0.046 \pm 0.027(1.7)$} & \scalebox{0.5}{$-0.02 \pm 0.026(-0.76)$} & \scalebox{0.5}{$-0.04 \pm 0.027(-1.5)$} & \scalebox{0.5}{$-0.012 \pm 0.027(-0.44)$} & \scalebox{0.5}{$0.13 \pm 0.027(4.8)$} & \scalebox{0.5}{$-0.04 \pm 0.027(-1.5)$} & \scalebox{0.5}{$-0.0056 \pm 0.027(-0.21)$} \\ \hline
  426. 6 & \scalebox{0.5}{$-0.013 \pm 0.026(-0.52)$} & \scalebox{0.5}{$0.041 \pm 0.027(1.5)$} & \scalebox{0.5}{$-0.033 \pm 0.028(-1.2)$} & \scalebox{0.5}{$-0.0019 \pm 0.027(-0.068)$} & \scalebox{0.5}{$0.15 \pm 0.027(5.4)$} & \scalebox{0.5}{$-0.034 \pm 0.027(-1.3)$} & \scalebox{0.5}{$-0.021 \pm 0.028(-0.75)$} \\ \hline
  427. %7 & \scalebox{0.5}{$0.25 \pm 0.027(9.1)$} & \scalebox{0.5}{$-0.27 \pm 0.027(-10)$} & \scalebox{0.5}{$-0.41 \pm 0.027(-16)$} & \scalebox{0.5}{$2.1 \pm 0.029(73)$} & \scalebox{0.5}{$0.65 \pm 0.027(24)$} & \scalebox{0.5}{$-0.19 \pm 0.028(-7)$} & \scalebox{0.5}{$-0.053 \pm 0.028(-1.9)$} \\ \hline
  428. 8 & \scalebox{0.5}{$0.039 \pm 0.026(1.5)$} & \scalebox{0.5}{$0.01 \pm 0.028(0.36)$} & \scalebox{0.5}{$-0.027 \pm 0.028(-0.96)$} & \scalebox{0.5}{$-0.024 \pm 0.026(-0.9)$} & \scalebox{0.5}{$0.12 \pm 0.027(4.3)$} & \scalebox{0.5}{$-0.092 \pm 0.026(-3.5)$} & \scalebox{0.5}{$-0.036 \pm 0.027(-1.3)$} \\ \hline
  429. 9 & \scalebox{0.5}{$-0.01 \pm 0.027(-0.38)$} & \scalebox{0.5}{$0.0024 \pm 0.027(0.09)$} & \scalebox{0.5}{$-0.018 \pm 0.026(-0.68)$} & \scalebox{0.5}{$0.022 \pm 0.028(0.79)$} & \scalebox{0.5}{$0.068 \pm 0.027(2.6)$} & \scalebox{0.5}{$-0.069 \pm 0.026(-2.6)$} & \scalebox{0.5}{$-0.078 \pm 0.026(-3)$} \\ \hline
  430. 10 & \scalebox{0.5}{$-0.017 \pm 0.027(-0.62)$} & \scalebox{0.5}{$-0.015 \pm 0.026(-0.57)$} & \scalebox{0.5}{$0.012 \pm 0.027(0.46)$} & \scalebox{0.5}{$-0.074 \pm 0.028(-2.6)$} & \scalebox{0.5}{$0.1 \pm 0.027(3.8)$} & \scalebox{0.5}{$-0.003 \pm 0.027(-0.11)$} & \scalebox{0.5}{$-0.043 \pm 0.029(-1.5)$} \\ \hline
  431. 11 & \scalebox{0.5}{$0.032 \pm 0.026(1.2)$} & \scalebox{0.5}{$0.024 \pm 0.026(0.91)$} & \scalebox{0.5}{$-0.04 \pm 0.026(-1.5)$} & \scalebox{0.5}{$-0.066 \pm 0.027(-2.5)$} & \scalebox{0.5}{$0.098 \pm 0.027(3.7)$} & \scalebox{0.5}{$-0.043 \pm 0.028(-1.6)$} & \scalebox{0.5}{$-0.0018 \pm 0.028(-0.064)$} \\ \hline
  432.  
  433.  
  434.  
  435. \hline
  436.  
  437.  
  438. \end{tabular}
  439.  
  440. \end{center}
  441.  
  442.  
  443. \end{tiny}
  444. \end{frame}
  445.  
  446. \begin{frame}\frametitle{Corr4- MM}
  447.  
  448.  
  449. \begin{tiny}
  450.  
  451. \begin{center}
  452. \begin{tabular}{ l l l l l l l l l }
  453. \hline
  454. \multicolumn{8}{c}{Mean of the pull} \\ \hline
  455. $q^2$ & $S_3$ & $S_4$ & $S_5$ & $S_{6s}$ & $S_{6c}$ & $S_7$ & $S_8$ \\ \hline \hline
  456.  
  457.  
  458. 0 & \scalebox{0.5}{$-0.019 \pm 0.026(-0.71)$} & \scalebox{0.5}{$0.048 \pm 0.027(1.8)$} & \scalebox{0.5}{$0.018 \pm 0.027(0.67)$} & \scalebox{0.5}{$0.059 \pm 0.027(2.2)$} & \scalebox{0.5}{$-0.015 \pm 0.028(-0.55)$} & \scalebox{0.5}{$0.061 \pm 0.027(2.2)$} & \scalebox{0.5}{$0.012 \pm 0.027(0.44)$} \\ \hline
  459. 1 & \scalebox{0.5}{$-0.014 \pm 0.028(-0.51)$} & \scalebox{0.5}{$0.043 \pm 0.027(1.6)$} & \scalebox{0.5}{$0.013 \pm 0.027(0.49)$} & \scalebox{0.5}{$0.024 \pm 0.028(0.86)$} & \scalebox{0.5}{$-0.038 \pm 0.028(-1.4)$} & \scalebox{0.5}{$0.024 \pm 0.026(0.94)$} & \scalebox{0.5}{$0.037 \pm 0.027(1.3)$} \\ \hline
  460. 2 & \scalebox{0.5}{$-0.03 \pm 0.027(-1.1)$} & \scalebox{0.5}{$-0.0021 \pm 0.027(-0.076)$} & \scalebox{0.5}{$-0.01 \pm 0.027(-0.39)$} & \scalebox{0.5}{$-0.017 \pm 0.027(-0.61)$} & \scalebox{0.5}{$-0.016 \pm 0.027(-0.58)$} & \scalebox{0.5}{$0.013 \pm 0.027(0.49)$} & \scalebox{0.5}{$0.027 \pm 0.028(0.98)$} \\ \hline
  461. 3 & \scalebox{0.5}{$-0.007 \pm 0.027(-0.26)$} & \scalebox{0.5}{$0.03 \pm 0.028(1.1)$} & \scalebox{0.5}{$-0.036 \pm 0.027(-1.3)$} & \scalebox{0.5}{$-0.074 \pm 0.027(-2.8)$} & \scalebox{0.5}{$0.041 \pm 0.027(1.5)$} & \scalebox{0.5}{$0.08 \pm 0.028(2.9)$} & \scalebox{0.5}{$-0.0083 \pm 0.027(-0.31)$} \\ \hline
  462. 4 & \scalebox{0.5}{$0.00089 \pm 0.029(0.031)$} & \scalebox{0.5}{$-0.021 \pm 0.027(-0.77)$} & \scalebox{0.5}{$0.0032 \pm 0.028(0.11)$} & \scalebox{0.5}{$0.0031 \pm 0.026(0.12)$} & \scalebox{0.5}{$0.012 \pm 0.027(0.44)$} & \scalebox{0.5}{$0.019 \pm 0.026(0.71)$} & \scalebox{0.5}{$0.034 \pm 0.027(1.3)$} \\ \hline
  463. 5 & \scalebox{0.5}{$0.044 \pm 0.028(1.6)$} & \scalebox{0.5}{$-0.022 \pm 0.026(-0.82)$} & \scalebox{0.5}{$-0.041 \pm 0.027(-1.5)$} & \scalebox{0.5}{$-0.014 \pm 0.027(-0.53)$} & \scalebox{0.5}{$-0.029 \pm 0.027(-1.1)$} & \scalebox{0.5}{$0.041 \pm 0.027(1.5)$} & \scalebox{0.5}{$0.042 \pm 0.028(1.5)$} \\ \hline
  464. 6 & \scalebox{0.5}{$-0.011 \pm 0.026(-0.42)$} & \scalebox{0.5}{$0.041 \pm 0.027(1.5)$} & \scalebox{0.5}{$-0.045 \pm 0.028(-1.6)$} & \scalebox{0.5}{$0.011 \pm 0.027(0.41)$} & \scalebox{0.5}{$-0.0089 \pm 0.027(-0.33)$} & \scalebox{0.5}{$0.067 \pm 0.027(2.5)$} & \scalebox{0.5}{$0.036 \pm 0.027(1.3)$} \\ \hline
  465. %7 & \scalebox{0.5}{$0.25 \pm 0.027(9.1)$} & \scalebox{0.5}{$-0.26 \pm 0.027(-9.9)$} & \scalebox{0.5}{$-0.42 \pm 0.027(-16)$} & \scalebox{0.5}{$2.1 \pm 0.029(73)$} & \scalebox{0.5}{$0.47 \pm 0.027(18)$} & \scalebox{0.5}{$-0.077 \pm 0.028(-2.8)$} & \scalebox{0.5}{$0.029 \pm 0.028(1)$} \\ \hline
  466. 8 & \scalebox{0.5}{$0.05 \pm 0.026(1.9)$} & \scalebox{0.5}{$0.0021 \pm 0.028(0.074)$} & \scalebox{0.5}{$-0.025 \pm 0.028(-0.91)$} & \scalebox{0.5}{$-0.023 \pm 0.026(-0.87)$} & \scalebox{0.5}{$-0.024 \pm 0.028(-0.85)$} & \scalebox{0.5}{$0.022 \pm 0.026(0.82)$} & \scalebox{0.5}{$0.026 \pm 0.026(0.98)$} \\ \hline
  467. 9 & \scalebox{0.5}{$-0.0064 \pm 0.027(-0.23)$} & \scalebox{0.5}{$0.012 \pm 0.027(0.44)$} & \scalebox{0.5}{$-0.0075 \pm 0.026(-0.28)$} & \scalebox{0.5}{$0.029 \pm 0.027(1.1)$} & \scalebox{0.5}{$-0.087 \pm 0.027(-3.2)$} & \scalebox{0.5}{$0.036 \pm 0.027(1.3)$} & \scalebox{0.5}{$-0.02 \pm 0.026(-0.76)$} \\ \hline
  468. 10 & \scalebox{0.5}{$-0.019 \pm 0.027(-0.71)$} & \scalebox{0.5}{$-0.0051 \pm 0.025(-0.2)$} & \scalebox{0.5}{$0.0081 \pm 0.026(0.31)$} & \scalebox{0.5}{$-0.077 \pm 0.028(-2.7)$} & \scalebox{0.5}{$-0.03 \pm 0.027(-1.1)$} & \scalebox{0.5}{$0.11 \pm 0.027(4.1)$} & \scalebox{0.5}{$0.013 \pm 0.028(0.46)$} \\ \hline
  469. 11 & \scalebox{0.5}{$0.027 \pm 0.026(1)$} & \scalebox{0.5}{$0.032 \pm 0.027(1.2)$} & \scalebox{0.5}{$-0.038 \pm 0.027(-1.4)$} & \scalebox{0.5}{$-0.048 \pm 0.026(-1.8)$} & \scalebox{0.5}{$-0.021 \pm 0.027(-0.77)$} & \scalebox{0.5}{$0.019 \pm 0.028(0.69)$} & \scalebox{0.5}{$0.035 \pm 0.027(1.3)$} \\ \hline
  470.  
  471.  
  472. \hline
  473.  
  474. \end{tabular}
  475.  
  476. \end{center}
  477.  
  478. \end{tiny}
  479. \end{frame}
  480.  
  481.  
  482.  
  483.  
  484. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  485. \section{Reverse Engineering Unfolding}
  486.  
  487. \begin{frame}\frametitle{Reverse Engineering- Corr1}
  488. \begin{itemize}
  489. \item Let's try to understand if we can understand why this happens:
  490. \item Let's calculate what should I expect with the unfolding
  491. \item This is up to normalization!
  492. \begin{itemize}
  493. \item $M_5=0.4 S_5 \to M_5=0.00512 S_3 + 0.4 S_5 - 0.002 S_7$
  494. \item $M_8=0.32 S_8 \to M_8=0.0016 S_4 + 0.00512 S_7 + 0.32 S_8$
  495. \item $M_7=0.4 S_7 \to M_7=0.002 S_5 + 0.4 S_7 + 0.00512 S_8$
  496. \item $M_3=0.32 S_3 \to M_3=0.32 S_3 - 0.0008 S_9$
  497. \end{itemize}
  498. \item The way you can look at this is that i just shown you how our unfolding matrix works like.
  499. \end{itemize}
  500.  
  501.  
  502.  
  503. \end{frame}
  504.  
  505. \begin{frame}\frametitle{Reverse Engineering- Corr2}
  506. \begin{itemize}
  507. \item Let's try to understand if we can understand why this happens:
  508. \item Let's calculate what should I expect with the unfolding
  509. \item This is up to normalization!
  510. \begin{itemize}
  511. \item $M_5=0.4 S_5 \to M_5= 0.4 S_5$
  512. \item $M_8=0.32 S_8 \to M_8=0.32 S_5$
  513. \item $M_7=0.4 S_7 \to M_7=0.4 S_7$
  514. \item $M_3=0.32 S_3 \to M_3=-0.0036 + 0.0012 Fl + 0.32 S_3$
  515. \end{itemize}
  516. \item The way you can look at this is that i just shown you how our unfolding matrix works like.
  517. \end{itemize}
  518.  
  519.  
  520.  
  521. \end{frame}
  522.  
  523.  
  524.  
  525.  
  526. \begin{frame}\frametitle{Summary}
  527. \begin{itemize}
  528. \item Developed a systematic way how to get Unfolding matrix
  529. \item Moments are resistant against variety of unfolding discrepancies.
  530. \item This might lead to reduced systematics in the future.
  531. \end{itemize}
  532.  
  533.  
  534.  
  535.  
  536. \end{frame}
  537.  
  538.  
  539.  
  540. \end{document}