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\title[Unfolding for counting experiments]{Unfolding for counting experiments}
\author{Marcin Chrz\k{a}szcz$^{1,2}$, Nicola Serra$^{1}$}
\institute{$^1$~University of Zurich,\\ $^2$~Institute of Nuclear Physics, Krakow}
\date{\today}
 
\begin{document}
% --------------------------- SLIDE --------------------------------------------
\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz}
% ------------------------------------------------------------------------------
% --------------------------- SLIDE --------------------------------------------
 
\begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
 
\center \includegraphics[width=0.8\paperwidth]{diagram.png}\\
 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
 
\center \includegraphics[width=0.8\paperwidth]{matrix.png}\\
 
\end{frame}
 
 
\section{Introduction}
\begin{frame}\frametitle{4D unfolding}
\begin{itemize}
\item Method of moments can use 2 different unfolding techniques.
\item One of them is Christophs 4D unfolding
\item Easy peasy:
\end{itemize}
\begin{equation}
\widehat{x}=\sum_i^n= \dfrac{f(\theta_{li}, \theta_{ki}, \phi_i)}{n} \rightarrow  \dfrac{f(\theta_{li}, \theta_{ki}, \phi_i) \times w_i}{\sum_j^n w_j}
\end{equation}
Very easy and has proved to work. See presentation \href{https://indico.cern.ch/event/290857/contribution/1/material/slides/0.pdf}{LINK}
\end{frame}
 
\begin{frame}\frametitle{Matrix unfolding}
\begin{itemize}
\item Our PDF is a vector in 8 dim space, where the dimensions are:\\
$f_{x}$\footnote{See 1st presentation on Method of moments. }
\item Our acceptance is a function:
\end{itemize}
\begin{equation}
\epsilon(\cos \theta_k, \cos \theta_l,\phi)
\end{equation}
\begin{itemize}
\item We assume it's a smooth function($C^{\infty}$).
\item Normal moments:
\end{itemize}
\begin{equation}
M_x=\int PDF \times f_x \rightarrow  \overline{M_x}=\int PDF \times f_x \times \epsilon(\cos \theta_k, \cos \theta_l,\phi)
\end{equation}
\begin{itemize}
\item Let's play Christophs trick:
\end{itemize}
\begin{equation}
\epsilon(\cos \theta_k, \cos \theta_l,\phi)=\sum A_{\alpha, \beta, \gamma}\cos^{\alpha} \theta_k \cos^{\beta} \theta_l  \phi^{\gamma}
\end{equation}
\end{frame}
 
 
 
 
\begin{frame}\frametitle{Matrix unfolding}
\begin{itemize}
\item Our Moments will become:
\end{itemize}
\small{
\begin{equation}
\int PDF \times f_x \times \epsilon(\cos \theta_k, \cos \theta_l,\phi) =\sum A_{\alpha, \beta, \gamma} \int PDF \times f_x \times \cos^{\alpha} \theta_k \cos^{\beta} \theta_l  \phi^{\gamma}
\end{equation}
}
\begin{itemize}
\item We just need to show that: 
\end{itemize}
 
\begin{equation}
\int PDF \times f_x \times \cos^{\alpha} \theta_k \cos^{\beta} \theta_l  \phi^{\gamma} = \sum_y  B_y \int PDF \times f_y = \sum_y B_y~M_y
\end{equation}
\begin{itemize}
\item This is a bit nasty but doable, using Mathematica. You just need to check 8 base functions.
\item As expected it's is linear terms.
\item I also checked this using analytical calculations.
\end{itemize}
 
\end{frame}
 
 
\begin{frame}\frametitle{Constructing Matrix unfolding}
\begin{itemize}
\item Ok, so we know that the matrix exists.
\item How ever we don't know explicate 
\end{itemize}
\small{
\begin{equation}
\epsilon(\cos \theta_k, \cos \theta_l,\phi) 
\end{equation}
}
\begin{itemize}
\item We don't, we just need to calculate matrix elements
\item Let's use PHSP MC.
\item Moments for PHSP MC are:\\
$v^{T}_{gen}=(2/3 ,0,0,0,0,0,0,0)$ 
\item After reconstruction we get:
$v^{T}_{rec}=( 0.7069,0.0077,-0.00236466,0.0005,0.0007,0.0011,0.0011,-0.0012)$
\end{itemize}
 
 
 
\end{frame}
 
\begin{frame}\frametitle{Constructing Matrix unfolding}
\begin{itemize}
\item We got first column of the unfolding matrix.
\end{itemize}
\small{
$ \begin{pmatrix}
  1.06  & \cdots & a_{1,8} \\
  0.01157  &  \cdots & a_{2,8} \\
 -0.003547  &  \ddots & \vdots  \\
 0.0007841 &  \ddots & \vdots  \\ 
  0.001126 &  \ddots & \vdots  \\ 
  0.001766 &  \ddots & \vdots  \\ 
   0.001664  &  \ddots & \vdots  \\ 
  -0.001937  &  \cdots & a_{8,8}
 \end{pmatrix}$
 
 
}
\begin{itemize}
\item How about the others?
\item We can reweight accordingly to $f_x$.
\end{itemize}
 
\end{frame}
 
\begin{frame}\frametitle{Constructing Matrix unfolding}
\begin{itemize}
\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) $
\item One can calculate on MC the reweighed moments in PHPS:
\end{itemize}
\begin{equation}
\int PDF*f_{S_3}=\dfrac{32}{225}
\end{equation}
\begin{itemize}
\item Our base vector now is:$v^{T}_{gen}=(0 ,\frac{32}{225},0,0,0,0,0,0)$ 
\item So lets see what do we get as reconstructed vector(after multiplying by $\frac{225}{32}$.
\small{$v^{T}_{rec}=(  0.042, 1.105,-0.005,0.003,-0.0023,-0.005,-0.005,-0.006)$ }
\end{itemize}
 
\end{frame}
   
 
\begin{frame}\frametitle{Constructing Matrix unfolding}
\begin{itemize}
\item Now the matrix looks like:
\end{itemize}
\small{
$ \begin{pmatrix}
  1.06 & 0.042  & \cdots & a_{1,8} \\
  0.01157 & 1.105  &  \cdots & a_{2,8} \\
 -0.003547 & -0.005  &  \ddots & \vdots  \\
 0.0007841 &-0.005  & \ddots & \vdots  \\ 
  0.001126 & 0.003 &\ddots & \vdots  \\ 
  0.001766 & -0.0023 &\ddots & \vdots  \\ 
   0.001664  & -0.005 &\ddots & \vdots  \\ 
  -0.001937  & -0.006 &\cdots & a_{8,8}
 \end{pmatrix}$
 
 
}
\begin{itemize}
\item The others go in the same way.
\item Repenting this exercise from $1^{st}$ year algebra we can get the full matrix
\end{itemize}
 
 
\end{frame}
\begin{frame}\frametitle{Constructing Matrix unfolding}
\begin{itemize}
\item The full transformation matrix from generator space to reconstructed space:
\end{itemize}
\tiny{
$ A_{gen\rightarrow reco}=\begin{pmatrix}
        1.06  &    0.0423 &  -0.0081 &   0.0022  &  0.0049 & 0.0037  &  0.0028 &  -0.0065 \\
 
    0.0115  &     1.105  & -0.0050 &   0.0027 &  -0.0018 & -0.0040 &-0.0054 &  -0.0065 \\
  -0.0035 &  -0.0050 &      0.981 &  0.0005 &  -0.0025 &  0.0002 &  -0.0037 &   -0.0048\\
  0.00078 &    0.0034 &  0.0006 &      1.002 &  -0.0032 & -0.0040  & 0.0003  &     0.0018\\
   0.001126  & -0.0023 &  -0.0032 &  -0.0032 &      1.055  &  0.001& -0.004 & 0.0023\\
   0.00176  & -0.0050 &  0.00036 &  -0.0040 &   0.0011 & 0.96 & -0.0057 & 0.0009 \\
   0.0016  & -0.005 &  -0.003 &  0.00029&  -0.003  &-0.004 &    0.9543 & 0.0000\\
  -0.0019 &  -0.0065 &   -0.004  &   0.001 &   0.0018  & 0.0007  &     0.000 &  1.098 \\
 
 
 
 
 
 \end{pmatrix}$
 
 
}
\begin{itemize}
\item Inverting the matrix is simple, and doable
\end{itemize}
\tiny{
$ A_{reco\rightarrow gen}=\begin{pmatrix}
     0.9434 &    -0.036 &   0.007&   -0.0020 &   -0.0044& -0.0038 &  -0.0030 &    0.0054\\ 
 -0.009 &     0.90 &   0.0045 &  -0.0024&   0.0016 &   0.003873 & 0.00527&     0.005 \\
   0.003 &   0.00454&      1.019 & -0.00058 &   0.0025& -0.000291 & 0.004 &  0.004 \\
-0.00071 &  -0.0030 & -0.0007 &     0.9977 &   0.0030 &   0.004206 &-0.0003 &   -0.0017  \\
  -0.001 &   0.0020 &   0.0031&   0.0030 &     0.9483 & -0.0010 &  0.004626 &  -0.0019 \\
 -0.001 &    0.004 & -0.0003 &   0.0042 &  -0.001087 &   1.037  &   0.0063  & -0.0009\\
   -0.0017 &   0.0053 &   0.0042 & -0.0002 &   0.00370& 0.0050 &  1.048 & 0.0000 \\ 
  0.0016&   0.0053&    0.00452 &  -0.001 &  -0.001582 &  -0.0007213 &0.000 &      0.9105 \\
 
 
 
 
 \end{pmatrix}$
}
 
\end{frame}
   
   
 
\begin{frame}\frametitle{Sensitivity to unknowns}
\begin{itemize}
\item We are unfolding based on MC.
\item There are MC/Data differences, which can have impact on the unfolding.
\end{itemize}
Let's put small modification:
\begin{equation}
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})
\end{equation}
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 :)
 
 
\end{frame}
 
\begin{frame}\frametitle{Corr1 functions}
\begin{columns}
\column{2in}
\includegraphics[width=\linewidth]{corr/Corr1_cosk.png}\\
\includegraphics[width=\linewidth]{corr/Corr1_cosl.png}\\
\column{2.5in}
$
corr1(\cos_l, \cos_k,\phi)= 1+ 0.032 \cos_l - 0.032  \cos_k + 0.01 \phi
$
\includegraphics[width=0.8\linewidth]{corr/Corr1_phi.png}\\
 
\end{columns}
 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Corr2 functions}
\begin{columns}
\column{2in}
\includegraphics[width=0.85\linewidth]{corr/corr21.png}\\
\includegraphics[width=0.85\linewidth]{corr/corr22.png}\\
\column{2.5in}
$
corr2(\cos_l, \cos_k,\phi)= -0.02 \cos_l^2 + 0.02 \cos_k^2  - 
 0.015 \phi^2+ 1
$
\includegraphics[width=0.75\linewidth]{corr/corr23.png}\\
 
\end{columns}
 
\end{frame}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Corr3 functions}
\begin{columns}
\column{2in}
\includegraphics[width=0.85\linewidth]{corr/corr31.png}\\
\includegraphics[width=0.85\linewidth]{corr/corr32.png}\\
\column{2.5in}
$
corr3(\cos_l, \cos_k,\phi)= 0.02 \cos_l \cos_k + 0.01 \cos_k \phi - 0.01 \phi \cos_l + 1
$
\includegraphics[width=0.75\linewidth]{corr/corr33.png}\\
 
\end{columns}
 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\frametitle{Corr4 functions}
\begin{columns}
\column{2in}
\includegraphics[width=0.85\linewidth]{corr/corr41.png}\\
\includegraphics[width=0.85\linewidth]{corr/corr42.png}\\
\column{2.5in}
$
corr3(\cos_l, \cos_k,\phi)= 0.01 \cos_k  \cos_l \phi + 1
$
\includegraphics[width=0.75\linewidth]{corr/corr43.png}\\
 
\end{columns}
 
\end{frame}
 
 
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
\begin{frame}\frametitle{Corr1- MM}
 
 
\begin{tiny}
 
\begin{center}
  \begin{tabular}{ l l l l l l l l l }
    \hline
      \multicolumn{8}{c}{Mean of the pull} \\     \hline
$q^2$ & $S_3$ & $S_4$ & $S_5$  &  $S_{6s}$  & $S_{6c}$ & $S_7$ & $S_8$  \\  \hline \hline
 
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
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
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
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
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
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
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
%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}{$-0.51 \pm 0.027(-19)$} & 
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
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
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
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 
 
 
 \hline
 
 
  \end{tabular}
 
\end{center}
 
 
\end{tiny}
\end{frame}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%                                                                 
 
 
\begin{frame}\frametitle{Corr2- MM}
 
 
\begin{tiny}
 
\begin{center}
  \begin{tabular}{ l l l l l l l l l }
    \hline
      \multicolumn{8}{c}{Mean of the pull} \\     \hline
$q^2$ & $S_3$ & $S_4$ & $S_5$  &  $S_{6s}$  & $S_{6c}$ & $S_7$ & $S_8$  \\  \hline \hline
 
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 
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
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
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
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
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
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
%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
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 
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
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
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
 
 
 
 
 
 
\hline
 
 
  \end{tabular}
 
\end{center}
 
 
\end{tiny}
\end{frame}
 
 
\begin{frame}\frametitle{Corr3- MM}
 
 
\begin{tiny}
 
\begin{center}
  \begin{tabular}{ l l l l l l l l l }
    \hline
      \multicolumn{8}{c}{Mean of the pull} \\     \hline
$q^2$ & $S_3$ & $S_4$ & $S_5$  &  $S_{6s}$  & $S_{6c}$ & $S_7$ & $S_8$  \\  \hline \hline
 
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 
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
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
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
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
 
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
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
%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
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 
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
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
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
 
 
 
\hline
 
 
  \end{tabular}
 
\end{center}
 
 
\end{tiny}
\end{frame}
 
\begin{frame}\frametitle{Corr4- MM}
 
 
\begin{tiny}
 
\begin{center}
  \begin{tabular}{ l l l l l l l l l }
    \hline
      \multicolumn{8}{c}{Mean of the pull} \\     \hline
$q^2$ & $S_3$ & $S_4$ & $S_5$  &  $S_{6s}$  & $S_{6c}$ & $S_7$ & $S_8$  \\  \hline \hline
 
 
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 
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
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
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
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
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
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
%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
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 
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
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
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
 
      
 
\hline
 
  \end{tabular}
 
\end{center}
 
\end{tiny}
\end{frame}
 
 
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Reverse Engineering Unfolding}
 
\begin{frame}\frametitle{Reverse Engineering- Corr1}
\begin{itemize}
\item Let's try to understand if we can understand why this happens:
\item Let's calculate what should I expect with the unfolding
\item This is up to normalization!
\begin{itemize}
\item $M_5=0.4 S_5 \to M_5=0.00512 S_3 + 0.4 S_5 - 0.002 S_7$
\item $M_8=0.32 S_8 \to M_8=0.0016 S_4 + 0.00512 S_7 + 0.32 S_8$
\item $M_7=0.4 S_7 \to M_7=0.002 S_5 + 0.4 S_7 + 0.00512 S_8$
\item $M_3=0.32 S_3 \to M_3=0.32 S_3 - 0.0008 S_9$
\end{itemize}
\item The way you can look at this is that i just shown you how our unfolding matrix works like.
\end{itemize}
 
 
 
\end{frame}
 
\begin{frame}\frametitle{Reverse Engineering- Corr2}
\begin{itemize}
\item Let's try to understand if we can understand why this happens:
\item Let's calculate what should I expect with the unfolding
\item This is up to normalization!
\begin{itemize}
\item $M_5=0.4 S_5 \to M_5= 0.4 S_5$
\item $M_8=0.32 S_8 \to M_8=0.32 S_5$
\item $M_7=0.4 S_7 \to M_7=0.4 S_7$
\item $M_3=0.32 S_3 \to M_3=-0.0036 + 0.0012 Fl + 0.32 S_3$
\end{itemize}
\item The way you can look at this is that i just shown you how our unfolding matrix works like.
\end{itemize}
 
 
 
\end{frame}
 
 
 
 
\begin{frame}\frametitle{Summary}
\begin{itemize}
\item Developed a systematic way how to get Unfolding matrix
\item Moments are resistant against variety of unfolding discrepancies.
\item This might lead to reduced systematics in the future.
\end{itemize}
 
 
 
 
\end{frame}
 
 
 
\end{document}