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\begin{document}
%\title{PREPARATION OF PAPER CONTRIBUTIONS FOR PUBLICATION IN THE CHEP06 PROCEEDINGS\thanks{Revised by Leena Chandran-Wadia, January 12, 2006}}
\title{SELF-ORGANIZED MAPS FOR TAGGING B JETS ASSOCIATED WITH HEAVY NEUTRAL MSSM HIGGS BOSONS}

%\author{J. Poole, C. Petit-Jean-Genaz, CERN, Geneva, Switzerland\\
%P. Lucas\thanks{lucas@fnal.gov}, FNAL, Batavia, IL 60510, USA}

\author{ A. Heikkinen\thanks{aatos.heikkinen@cern.ch}, S. Lehti, Helsinki Institute of Physics, Helsinki, Finland}

\maketitle

\begin{abstract}
B tagging is an important tool for separating the LHC Higgs events 
with associated b jets from the Drell-Yan background~\cite{segneri02}.
We extend standard neural network (NN) approach 
using multilayer perceptron in b tagging~\cite{heikkinen05} 
to include self-organizing feature maps. 
We demonstrate the use of the self-organizing maps (SOM\_PAK program package) and 
the learning vector quantization (LVQ\_PAK) ~\cite{kohonenPak}. 
A background discriminating power of these NN tools are compared 
with standard tagging algorithms.
\end{abstract}

\section{INTRODUCTION}

In MSSM the heavy neutral Higgs boson production in association with two b quarks is the 
dominant Higgs boson production mechanism at large values of tan$\beta$ at LHC. 
These associated b jets can be used to extract the Higgs events from the
Drell-Yan $\rm Z/\gamma^*$ background, 
for which the associated jets are mostly light quark and gluon jets. 

\subsection{b-tagging}
In standard methodology a jet can be identified as a b jet using lifetime 
based tagging algorithm, which relies on displaced  secondary vertices and 
track impact parameter (IP). Impact parameter is the closest approach of
the track trajectory to the primary vertex.
Figure \ref{vertexCloseup} demonstrates the case in CMS detector at Large Hadron Collider.

For a review of the main algorithms for 
inclusive b-tagging based on track IP and secondary vertex, see Refs. ~\cite{segneri02} and ~\cite{weiser06}.


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\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{vertexCloseup.eps}
\caption{A displaced secondary vertex in a $\rm b\bar{\rm b}H$ event with
$\rm H\rightarrow\tau\tau$ in CMS. The second b jet is not reconstructed due
low jet energy and low track multiplicity. 
%of CMS event as simulated with ORCA~\cite{segneri02} package. A b jet with displaced secondary vertice is shown.
}
\label{vertexCloseup}
\end{figure}


\section{Self-organizing maps (SOMs)}

In comparison to quite common multi layer perceptrons (MLPs) technique (See for example Refs. ~\cite{zimmermann04a, hultqvist95a, iashvili96, mjahed05}.),
only few authors have reported on SOM (also known as Kohonen network) use in separation of background from signal, 
though some promising results have been achieved ~\cite{deAngelis04, becks99a, lange99a, lang98a}. 
Yet SOM is can provide computationally more simple algorithm, with learning rate  faster than what MLPs have.

%only a few papers has been written on unsupervised in high energy physics field ::: ~\cite{deAngelis04}.

The most popular unsupervised neural network algorithm SOM, 
defines a mapping from n-dimensional input data space onto a regular two-dimensional array of neurons~\cite{haykin99, kohonen95}:

\begin{Itemize}
\item Every neuron of the map is associated with an n-dimensional reference vector. 
\item The neurons of the map are connected to adjacent neurons by a neighborhood relation, which dictates the topology of the map. 

\item During the unsupervised training phase, the SOM forms an elastic net that folds onto the "cloud" formed by 
input data and  approximates the density of the data.
\end{Itemize}

\subsection{Competitive process}
The SOM defines a mapping from the input data space 
$${\bf x} = [x_1, x_2, ..., x_m]^T$$ 
onto a regular two-dimensional array of nodes.
The synaptic weight vector 
$${\bf w}_j = [w_{j1}, w_{j2}, ..., w_{jm}]^T ,  ~j = 1, 2, ..., l$$ 
of each neuron $j$ has the same dimension as the input space; $l$ is a total number of neurons. 


Selecting the neuron with the largest inner product ${\bf w}_j^T{\bf x}$, is mathematically equivalent 
to minimizing the Euclidean distance between the input vectors ${\bf x}$ and ${\bf w}_j$.
Thus, the  winning neuron $c$ is defined as:
$$c = arg ~min_j ||{\bf x} - {\bf w}_j||$$

Essentially this sums up the competition process among the neurons, where the best-matching node locates the center
of a topological neighborhood~\cite{haykin99}.

\subsection{Adaptive process}
During the learning, those nodes that are topographically close to a certain distance 
will activate each other to learn from the same input.
Using discrete-time formalism, weight vector at time $t$ is written as ${\bf w}_j(t)$, 
and updated weight vector is defined as:
$${\bf w}_j(t+1) = {\bf w}_j(t)+{\bf h}_{jc}(t)[{\bf x}-{\bf w}_j(t)]$$
where ${\bf h}_{jc}(t)$ is neighborhood kernel. For details see Ref. \cite{zurada03a}.

\section{A SOM APPROACH TO B-TAGGING}

\subsection{Event data}
In our SOM approach we feed SOM network with the same events and the same variables as used in the traditional track counting algorithm: 

\begin{Itemize}
\item Number of tracks in the jet cone

\item Impact parameters and related IP significances for three leading tracks. (See Fig. \ref{variables}.) 
\end{Itemize}

Figure \ref{vertexProjections} shows an example of CMS event used in this study.

\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{vertexProjections.eps}
\caption{Successfully reconstructed  jet can be identified 
as a b jets using lifetime based tagging algorithm, 
which relies on displaced secondary vertices. 
Two leptons leptons are presented by straight lines.}
\label{vertexProjections}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{variables.eps}
\caption{An example of variables used in the SOM teaching. A leading track IP distribution for a signal  (left) and background events and a significance of a  leading track IP (right).}
\label{variables}
\end{figure}


\subsection{SOM configuration and Learning Vector Quantization LVQ}
We used SOM\_PAK ~\cite{kohonenPak} tool to analyze data created in CMS ORCA~\cite{segneri02} simulation package, 
using full simulation with track and jet reconstruction.


In our hybrid approach, after the self-organizing feature map, 
the actual classification and feature map tuning (For details see Ref. ~\cite{haykin99}.) was performed  
using a learning vector quantizer tool LVQ\_PAK ~\cite{kohonenPak}. 
Details of this type of winner-takes-all algorithm are available elsewhere in
the literature~\cite{zurada03a, biehl06}.


We started learning with the optimized LVQ1 algorithm, 
which has a very fast convergence. 
Some performance improvement was achieved with subsequent use of LVQ2.1 algorithm.


\section{RESULTS}
The b tagging efficiency with SOM was found to be 73~\% with 
11~\% mistagging rate.
%SOM was able to do b-tagging with 73~\% efficiency and 11~\% mistagging rate.
We were able to filter 45~\% of the background events with
%Also it filtered 45~\% background events with 
0.2\% misclassification probability for the signal.
Subsequent tuning of SOM with LVQ\_PAK-tool provided 1.5~\% improvement in 
tagging efficiency.
%divide to signal and background regions is visualized in Fig. \ref{sideBySideSB}. 
In Figs. \ref{signalArea} and \ref{bgArea}, which visualize SOM activation 
with test data, a clear separation
%divide 
to signal and background regions is seen.
% in Figs. \ref{signalArea} and 
%\ref{bgArea} which visualize SOM activation with test data. 
These results can be compared with typical counting algorithms performance 
35~\% efficiency with 1~\% mistagging propability. 

%\begin{figure*}[t]
\begin{figure}[htb]
\centering
%\includegraphics*[width=165mm]{sideBySideSB.eps}
\includegraphics*[width=80mm]{../../../images/procSignalArea.eps}
\caption{Number of signal events associated to winning node, while testing mapping of 15x15 node SOM performance.}

%\caption{Number of background (right)  events associated to winning node, while testing mapping of 15x15 node SOM performance. A clear divide to signal and background regions is seen.}

\label{signalArea}
%\end{figure*}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{procBackgroundArea.eps}
\caption{Number of background events associated to winning node, while testing mapping of 15x15 node SOM performance. }
\label{bgArea}
\end{figure}


%margin of the column (see Eq.~\ref{eq:units}). Units should be written
%\begin{equation}\label{eq:units}
%C_B={q^3\over 3\epsilon_0 mc}=3.54\,\hbox{$\mu$eV/T}
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\section{CONCLUSION}

We have shown that  unsupervised classification can be utilized successfully in 
b-tagging problems. 
In our study the self-organizing maps are able to separate the Higgs signal 
from the background, based on CMS Monte Carlo data.

We consider HEP as a promising field where the potential of self-organizing 
networks has not yet been fully exploited.

Since the optimization of free parameters $\epsilon$ and $\delta$ and 
their time dependence can be very time consuming,
our future work will consist of inclusion of growing SOM using a gravitational 
algorithm ~\cite{lange99a}.
We are also planning to study very promising AdaBoost-algorithm  as an 
alternative to artificial neural networks 
for the Higgs boson identification ~\cite{roe05a}.

\begin{thebibliography}{9}   % Use for  1-9  references
%\begin{thebibliography}{99} % Use for 10-99 references

\bibitem{segneri02} G.~Segneri and F.~Palla, ``Lifetime based b-tagging with CMS'', CMS NOTE 2002/046, November 26, 2002.

\bibitem{heikkinen05} A. Heikkinen and S. Lehti, ``Tagging b jets associated with heavy neutral MSSM Higgs bosons''. 
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%\bibitem{heikkinen05} A. Heikkinen and S. Lehti, ``Tagging b jets associated with heavy neutral MSSM Higgs bosons''. 
%Proceedings of ACAT 2005, May 22 - 27, DESY, Zeuthen, Germany.
%Tagging b jets associated with heavy neutral MSSM Higgs bosons  %Gâ��%@ ARTICLE
%Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, 
%Detectors and Associated Equipment, Volume 559, Issue 1, 1 April 2006, Pages 195-198
%A. Heikkinen and S. Lehti

\bibitem{kohonenPak} %http://www.cis.hut.fi/research/som_lvq_pak.shtml
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    T.~Kohonen {\it et. al.}, LVQ\_PAK: The Learning Vector Quantization Program Package. Technical Report A30, Helsinki University of Technology, Laboratory of Computer and Information Science, FIN-02150 Espoo, Finland, 1996.


\bibitem{weiser06} C.~Weiser, ``A Combined Secondary Vertex Based B-Tagging Algorithm in CMS'', CMS NOTE 2006/014, January 25, 2006.



%\bibitem{chep04-ref}
%http://www.chep2004.org.


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\bibitem{deAngelis04} A. De Angelis {\it et al.}, ``Self-organizing networks for classification: developing applications to science analysis for astroparticle physic'', Physics A 338 (2004) 50-53.

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\end{thebibliography}

\end{document}
