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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}
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\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz}
% ------------------------------------------------------------------------------
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\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}
mchrzasz