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\title[Extracting angular observables with Method of Moments]{Extracting angular observables\\ with Method of Moments}
\author{Marcin Chrz\k{a}szcz$^{1}$ \\in collaboration with \\Frederik Beaujean, Nicola Serra and Danny van Dyk,\\{~}\\ based on \href{http://arxiv.org/abs/1503.04100}{\textit{arXiv:1503.04100}} }
\institute{$^1$~University of Zurich}
\date{\today}

\begin{document}
% --------------------------- SLIDE --------------------------------------------
\frame[plain]{\titlepage}
\author{Marcin Chrz\k{a}szcz{~}}
\institute{(UZH)}
% ------------------------------------------------------------------------------
% --------------------------- SLIDE --------------------------------------------
\tableofcontents

\placelogotrue
\section{Motivation}
\begin{frame}\frametitle{Motivation}

Likelihood(LL) fits even though widely used suffer from couple of draw backs:
\begin{enumerate}
\item In case of small number events LL fits suffer from convergence problems. This behaviour is well known and was observed several times in toys when we done $\PB \to \PKstar \Pmu \Pmu$.
\item LL can exhibit a bias when underlying physics model is not well known, incomplete or mismodeled.
\item The LL have problems converging when parameters of the \pdf are close to their physical boundaries, so-called ''boundary problem''
\item Accessing uncertainty in LL in some cases requires application of computationally expensive Feldman-Cousins method.
\end{enumerate}



\end{frame}



\begin{frame}\frametitle{Method of Moments}
\begin{columns}
\column{0.05in}{~}
\column{2.2in}
\begin{center} MoM solves the above problems:\end{center}

\column{2.in}
\only<3>{
\begin{center} Drawback:\end{center}
}
\end{columns}



\begin{columns}
\column{0.05in}{~}
\column{2.2in}
\only<1>{
%\begin{center} MoM solves the above problems:\end{center}
\begin{exampleblock}{Advantages of MoM}
  \begin{itemize}
  \item Probability distribution function rapidity converges towards the Gaussian distribution. 
  \item MoM gives an unbias result even with small data sample.
  \item Insensitive to large class of remodelling of physics models.
  \item Is completely insensitive to boundary problems.
  \end{itemize}
\end{exampleblock}
}
\only<2,3>{
%\begin{center} MoM solves the above problems:\end{center}
\begin{exampleblock}{Advantages of MoM}
  \begin{itemize}
  \item Each observable can be determined separately from other.
  \item Uncertainly follows perfectly $1/\sqrt{N}$ scaling.
  \end{itemize}
\end{exampleblock}
}
\column{2.in}
\only<3>{

%\begin{center} Drawback:\end{center}
\begin{alertblock}{Advantages of MoM}
  \begin{itemize}
  \item Estimated uncertainty in MoM is larger then the ones from LL.
  \end{itemize}
\end{alertblock}
}

\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Method of Moments}
\begin{frame}\frametitle{Introduction to MoM}
Let us a define a probability density function \pdf of a decay:
\begin{align}
P(\nuvec, \thvec) \equiv \sum_i S_i(\nuvec) \times f_i(\thvec)
\end{align}
Let's assume further that there exist a dual basis: $\lbrace f_i(\thvec) \rbrace$, $\{\dual{f}_i(\thvec)\}$ that the orthogonality relation is valid:
\begin{equation}                                                                                                                                                                                                                              
    \label{eq:def-ortho-rel}                                                                                                                                                                                                                  
    \int_\Omega \rmdx{\vec{\theta}} \dual{f}_i(\thvec) f_j(\thvec)  = \delta_{ij}                                                                                                                                                             
\end{equation}    
Since we want to use MoM to extract angular observables it's normal to work with Legendre polynomials. In this case we can find self-dual basis: 
\begin{equation}
\forall_i \dual{f}_i = f_i~,
\end{equation}

just by applying the ansatz: $\dual{f}_i=\sum_i a_{ij} f_j$.

\end{frame}

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\begin{frame}\frametitle{Determination of angular observables}
Thanks to the orthonormality relation Eq.~\ref{eq:def-ortho-rel} one can calculate the $S_i(\nuvec)$ just by doing the integration:
\begin{align}
S_i(\nuvec)=\int_\Omega d \thvec  P(\nuvec, \thvec) \dual{f}_i(\thvec) 
\end{align}
\pause
We also need to integrate out the $\nuvec$ dependence:
\begin{align}\label{eq:canonical}
\langle S_i \rangle= \int_\Theta d \nuvec  \int_\Omega d \thvec P(\nuvec, \thvec) \dual{f}_i(\thvec) 
\end{align}
\pause
MoM is basically performing integration in~Eq.~\ref{eq:canonical} using MC method:
\begin{align*}
E[S_i] \to \widehat{E[S_i]}=\dfrac{1}{N}\sum_{k=1}^{N} \dual{f}(x_k)
\end{align*}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}\frametitle{Uncertainty estimation}
MoM provides also a very fast and easy way of estimating the statistical uncertainty:
\begin{align}
\sigma (S_i)= \sqrt{\dfrac{1}{N-1}\sum_{k=1}^N ( \dual{f}_i(x_k) - \widehat{S_i} )^2   } 
\end{align}
and the covariance:
\begin{equation}
\mathrm{Cov}  [S_i, S_j]=\dfrac{1}{N-1} \sum_{k=1}^N [ \widehat{S_i} - \dual{f}_i(x_k) ][ \widehat{S_j} - \dual{f}_j(x_k) ]
\end{equation}
\pause
Thanks to the CLT both equations are satisfied.



\end{frame}
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\section{Systematic uncertainties}
\begin{frame}\frametitle{Partial Waves mismodeling}


\end{frame}


\begin{frame}\frametitle{Detector effects}



\end{frame}







              
\end{document}