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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 --------------------------------------------
 
\institute{~(UZH, IFJ)}
 
\section{Introduction}
 
 
\begin{frame}\frametitle{Reminder 1 - Constructing Matrix unfolding}
\begin{itemize}
\item We don't know explicate 
\item I have proven some time ago that the matrix exist
\end{itemize}
\small{
\begin{equation}
\epsilon(\cos \theta_k, \cos \theta_l,\phi) 
\end{equation}
}
\begin{itemize}
\item I have proven some time ago that the matrix exist
\item Now a systemic way to produce it. 
\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(full $q^2$ range):
$v^{T}_{rec}=( 0.7069,0.0077,-0.00236466,0.0005,0.0007,0.0011,0.0011,-0.0012)$
\end{itemize}
 
 
 
\end{frame}
 
\begin{frame}\frametitle{Reminder 2 - Constructing Matrix unfolding}
\begin{itemize}
\item We got first column of the unfolding matrix $(\dfrac{3}{2} v_{gen})$.
\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{Reminder 3 - 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{$v^{T}_{rec}=(  0.042, 1.105,-0.005,0.003,-0.0023,-0.005,-0.005,-0.006)$ }
\item Please notice that weights are negative, but this is not a problem for the mean.
\item Also we are avoiding the negative PDF problem :)
\end{itemize}
 
\end{frame}
   
 
 
\begin{frame}\frametitle{Reminder 4 - 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{Reminder 5}
For now:
\begin{itemize}
\item We have proven that there has to exists unfolding matrix.
\item Shown how to construct transformation matrix: $Gen \to Reco$.
\item Inverting it we can have transformation matrix of $Reco \to Gen$.
\item For details: \href{https://indico.cern.ch/event/316905/session/1/contribution/18/material/slides/0.pdf}{LINK}
 
\end{itemize}
 
What is missing?
\begin{columns}
\column{1in}
\begin{enumerate}
\item ERROR!
\end{enumerate}
\column{4in}
\includegraphics[width=0.8\textwidth]{err.jpg}\\
\end{columns}
 
\end{frame}
 
 
 
 
 
 
\begin{frame}\frametitle{How to?}
\begin{itemize}
\item So lets say that transformation matrix:$Gen \to Reco$ is $\epsilon_{i,j}$.
\item Each element has an error:$\delta \epsilon_{i,j}$.
\item Then we can calculate the matrix: $\epsilon_{i,j}^{-1}$(assuming it exists).
\item The million dollar question is what is the error on inverted matrix?
\end{itemize}
 
\end{frame}
 
 
 
\begin{frame}\frametitle{Answer to 1M dolar quesion}
\only<1>{
\begin{itemize}
\item One can toy it.
\item But toying is good for kids and Frequentist. 
\end{itemize}
 
}
\only<2>{
 
\begin{itemize}
\item One can toy it.
\item But toying is good for kids and Frequentist. 
\end{itemize}
 
\begin{itemize}
\item Solution comes from $\tau$ physics :) \href{http://arxiv.org/abs/hep-ex/9909031}{hep-ex/9909031}
\end{itemize}
\begin{itemize}
\item One can derive(prove in the paper) the general equation:
\end{itemize}
\begin{equation}
\delta \epsilon^{-1}_{\alpha ~ \beta}= [\epsilon^{-1}]^2_{\alpha  i}[\delta \epsilon ]^2_{ij} [\epsilon^{-1}]^2_{j \beta}
\end{equation}
}
 
 
\end{frame}
 
 
 
\begin{frame}\frametitle{Matrix, $1.1-2~GeV$}
\tiny{                                                                                       
$ A_{reco\rightarrow gen}=\begin{pmatrix}                                                    
 0.9519 & -0.02665 & -0.01432 & 0.002356 & 0.02539 & 0.009878 & -0.01551 & -0.01874 \\
-0.006272 & 0.8122 & -0.00351 & -0.00719 & 0.003585 & 6.784e-05 & 0.02445 & 0.008515 \\ 
-0.005315 & -0.003716 & 1.048 & 0.01242 & 0.01209 & -0.01478 & -0.001956 & 0.01429  \\
0.003237 & -0.007177 & 0.01533 & 0.9184 & -0.007548 & -0.0009818 & -0.01874 & 0.009407  \\ 
0.01002 & 0.004084 & 0.01391 & -0.006509 & 1.194 & -0.006516 & 0.001536 & -0.02882  \\
0.002695 & -0.001042 & -0.01721 & -0.001842 & -0.005643 & 0.9264 & 0.02106 & 0.006755  \\
-0.004736 & 0.02346 & -0.002335 & -0.01446 & 0.001169 & 0.01697 & 1.072 & -0.003191 \\
-0.004157 & 0.007576 & 0.01377 & 0.008058 & -0.02219 & 0.005354 & -0.0008608 & 0.8304  
 
                                                                              
 \end{pmatrix}$                                                                              
}                                                                                            
{~}\\{~}\\{~}\\
\tiny{                                                                                       
$ \delta A_{reco\rightarrow gen}=\begin{pmatrix}                                                    
0.005202 & 0.01911 & 0.03258 & 0.02103 & 0.02252 & 0.02145 & 0.03366 & 0.01948 \\
0.006648 & 0.04654 & 0.03227 & 0.02451 & 0.03602 & 0.02464 & 0.03298 & 0.03397 \\
0.007557 & 0.03197 & 0.07845 & 0.04272 & 0.04744 & 0.03057 & 0.05698 & 0.03287 \\
0.007902 & 0.03885 & 0.0678 & 0.04839 & 0.0384 & 0.03464 & 0.04925 & 0.03989 \\
0.009015 & 0.04122 & 0.06374 & 0.03254 & 0.07349 & 0.03269 & 0.0649 & 0.04202 \\ 
0.007939 & 0.0389 & 0.04793 & 0.03433 & 0.03828 & 0.04937 & 0.06985 & 0.04023 \\
0.007651 & 0.03234 & 0.05611 & 0.03062 & 0.04776 & 0.04388 & 0.08157 & 0.03342 \\ 
0.006719 & 0.03345 & 0.03868 & 0.02953 & 0.03633 & 0.03002 & 0.03989 & 0.04827 
 
 
                                                                              
 \end{pmatrix}$                                                                              
}                   
 
\end{frame}
\begin{frame}\frametitle{What did go wrong?}
\begin{itemize}
\item The errors are $2-3\%$, which is very worrying.
\item WG got very worried what is going on with the errors :(
\item Started debugging.
\item After sleeping with the problem found a stupid:
\end{itemize}                                                                        
\textbf{  for(int i=0;i $<$  entries/10;++i)   }
\begin{itemize}
\item Ok, I am an idiot, and used $10\%$ of statistics.
\end{itemize} 
\end{frame}
 
\begin{frame}\frametitle{What did go wrong 2 ?}
\begin{itemize}
\item The errors are tricky. When you re-weight you have negative weights. 
\item So I change the normal error
\end{itemize}
\begin{equation}
\Hat\sigma^2 = \dfrac{\sum w_i}{(\sum w_i)^2 - \sum w_i^2} 
                \sum w_i (x_i - \Hat\mu)^2
\end{equation}
\begin{itemize}
\item to:
\end{itemize} 
\begin{equation}
\Hat\sigma^2 = \dfrac{\sum  |w_i|  }{(\sum |w_i|)^2 - \sum w_i^2} 
                \sum |w_i| (x_i - \Hat\mu)^2
\end{equation}
\begin{itemize}
\item And this I am not $100\%$ sure if I is ok =(
\end{itemize} 
 
\end{frame}
\begin{frame}\frametitle{What did go wrong 3 ?}
\begin{itemize}
\item There is a hack of this method:
\item "You can cheat on your gf, you can cheat on tax, but you can't cheat on $\sqrt{n}$ "\footnote{All rights reserved! }.
\end{itemize} 
\begin{center}
\includegraphics[width=0.5\textwidth]{Q2_5_6_S5.png}\\
\end{center}
\begin{itemize}
\item We can use this:
\item Divide the MC in 10. Then calculate the variance of each matrix element. And divide/multiply by $\sqrt{10}$ and see if the errors are ok.
\end{itemize} 
 
 
\end{frame}
 
\begin{frame}\frametitle{What did go wrong 3 ?}
\tiny{    OLD (can be wrong): \\                                                                                   
$ \delta A_{gen\rightarrow reco}=\begin{pmatrix}                                                    
  0.005477  &   0.02348  &   0.03125  &   0.02305 &    0.01871  & 0.02307  &   0.03124   &  0.02339 \\
  0.008142  &   0.06734  &   0.03621  &   0.03126 &     0.0352  &  0.03131 &    0.03624  &   0.04767 \\
  0.007168  &    0.0359  &   0.06856  &    0.0423 &    0.03619  & 0.02995  &   0.04856   &  0.03585 \\
  0.008573  &   0.04966  &   0.06736  &   0.05471 &    0.03332 & 0.03886  &   0.04784    & 0.04973 \\
  0.007599  &   0.04063  &   0.04926  &   0.02847 &    0.04998 & 0.02841  &   0.04923    & 0.04059 \\
  0.008582  &   0.04977  &   0.04768  &   0.03878 &    0.03323 & 0.05499  &    0.0676    & 0.04974 \\
  0.007136  &   0.03571  &   0.04833  &   0.02987 &      0.036 & 0.04225  &   0.06843    &  0.0358  \\
  0.008162  &   0.04782  &   0.04294  &   0.03731 &    0.03527 &  0.03738  &   0.04306   &  0.06736 
 
                                                                              
 \end{pmatrix}$                                                                              
}           
 
\tiny{    New: \\                                                                                   
$ \delta A_{gen\rightarrow reco}=\begin{pmatrix}                                                    
  0.006659   &   0.0299  &   0.02207  &   0.01802  &   0.02657  &  0.02196  &   0.02851  &   0.02507 \\
   0.00708   &  0.02046 &   0.007998  &    0.0133  &  0.008828  &  0.01236  &   0.01505  &    0.0149\\
  0.008469   &  0.00845 &    0.01806  &   0.01442  &  0.009856  &  0.008895 &    0.01389 &    0.01155\\
  0.008938   &  0.01569 &    0.01798  &   0.01801  &  0.009195  &  0.01097  &   0.01108  &   0.02068\\
  0.007867   &   0.0109 &    0.01248  &    0.0088  &   0.01104  &  0.0114  &   0.01256   &  0.01097\\
  0.008078   &  0.01582 &    0.01117  &   0.01093  &   0.01135  &  0.01215 &    0.02122  &   0.01774 \\
  0.008368   &  0.01521 &    0.01391  &  0.008972  &  0.009797  &  0.01702 &     0.0147  &   0.01086\\
  0.005745   &  0.01561 &     0.0114  &   0.01649  &  0.008631 &  0.01373  &   0.01051   &  0.01792 
                                                                              
 \end{pmatrix}$                                                                              
}           
 
 
 
 
\end{frame}
 
 
 
 
 
\begin{frame}\frametitle{Summary}
\begin{itemize}
\item I really fu.. this thing ...
\item No coding after 3 am form now!
\end{itemize} 
 
\includegraphics[width=0.5\textwidth]{code.png}\\
 
\end{frame}
 
 
 
 
 
 
 
 
\end{document}