1. Search for point-like source in ANTARES
Juan Antonio Aguilar Sánchez
IFIC (Instituto de Física Corpuscular) CSIC-Universitat de València, Spain
on behalf of the ANTARES collaboration

2. ANTARES: See talk from A. Kouchner
Motivation
Scientific scope of a Cherenkov Neutrino Telescope:
Search for point-like sources is one of the main motivations to build a Neutrino Telescope ?
~MeV
GeV-100 GeV
GeV-TeV
TeV-PeV
PeV-EeV
> EeV
ANTARES: See talk from A. Kouchner
Background: Mostly Atmospheric neutrinos
ANTARES:
Very good angular resolution
Galactic Centre visible 63 % of time

3. Methods for the search of point-like sources
Different methods have been developed within ANTARES collaboration for the search of point-like sources:
Grid/Cluster
ML Ratio
EM algorithm
Unbinned techniques
Binned techniques
Signal-like
Background-like
They are more powerful than binned techniques.
They use the precise configuration of the events.
No optimization is needed in unbinned methods
They require more CPU time and Monte Carlo experiments
They are well-known and very robust
They do not have a strong dependence on the detector performances
They need a bin/cone optimization
Significances are easily computed and analytically derived.

4. Grid and Cluster Methods
Sky is divided in a grid of squared bins or cones around each sample event.
The optimum size of the bins/cones is calculated for maximum sensitivity, with the additional criteria of having the same number of background events per bin/cone (uniform sensitivity)
Significance: Si=log10(Pi)
Grid method
Grid/Cluster
ML Ratio
EM algorithm
Pi is the probability for the background to produce Ni or more events if there are Ntotal events in the declination band d RA
Pi is the probability of the background to produce the observed number of events N0 or more (up to the maximum number Ntotal).  is each element of the set CnNtotal of combinations of Ntotal elements in groups of n elements.
Cluster method

5. Likelihood Ratio Method
Try to develop a method that uses all available information:
1.- Event distribution
2.-Energy information
3.-Energy dependence of the angular resolution
Two hypotheses:
H0: only atmospheric neutrinos
H1: background and some signal
As a discriminator observable (test statistic) the likelihood ratio is used:
likelihood if there is signal+bg
likelihood for bg-only
Likelihood is expressed as a sum over the events
Grid/Cluster
ML Ratio
EM algorithm
Background density:
point spread function
Pdf muon energy reconstruction
detector acceptance
Knowledge of detector:
Signal density:
Parameters to fit:
flux magnitude
source position
flux spectral index

6. The EM algorithm Grid/Cluster ML Ratio EM algorithm
The EM method is a pattern recognition algorithm that analytically maximizes the likelihood in finite mixture problems, which are described by different density components (pdf) as:
position of event: x = (αRA, δ)
signal: αRA, 
bg: only 
Point-like
sources
Parameters to fit:
source position
Gaussian width
Grid/Cluster
ML Ratio
EM algorithm
pdf
proportion of signal and background
Selected model:
The background pdf is extracted from MC data or real RA-scrambled data when available
Signal pdf model is selected to be 2D-Gaussians

7. Easily differentiable!
General procedure
The idea is to assume that the set of observations forms a set of incomplete data vectors. The unknown information is whether the observed event belongs to a component or another.
COMPLETE data set
INCOMPLETE data set
The vector zi is a class indicator that indicates if the event i belongs to the background or the source.
Easily differentiable!
E-Step (Expectation-step):
Start with a set of initial parameters Ψ(m) = {π1,π2,µ,Σ}
Expectation of the complete data log-likelihood, conditional on the observed data {x}
M-Step (Maximization-step):
Find Ψ = Ψ(m + 1) that maximizes Q(Ψ, Ψ(m))
Grid/Cluster
ML Ratio
EM algorithm
L(Ψ)
Q(Ψ,Ψ(m))+hm
Ψ(m)
Ψ(m+1)
Ψ(m+2) Ψ
Q(Ψ,Ψ(m+1))+hm+1
Successive maximizations of the function Q(Ψ,Ψ(m)) lead to the maximization of the log-likelihood

8. Searching procedure Grid/Cluster ML Ratio EM algorithm
Pre-clustering algorithm
Initial values Y(m):
-m cluster barycenter
-s cluster size
-pS cluster elements
As a discriminator observable we use the Bayesian Information Criterion (BIC) used in a frequentist fashion:
E-step:
Compute Q(Y,Y(m))
Background like
Signal like
BIC3σ
BIC5σ
m = m +1
M-step:
Find Y* = arg max Q(Y,Y(m))
10000 experiments
Grid/Cluster
ML Ratio
EM algorithm
Y(m+1) = Y* No
The discovery power of the test (or discovery potential) is the percentage of success in detecting a source over the background
where Nσ = 3σ, 5σ
Q(Y(m+1),Y(m)) –
Q(Y(m),Y(m-1)) ≤ x
Yes
YML = Y(m+1)

9. Results and Comparison
Results with different neutrino Monte Carlo and muon track reconstruction strategies. Direct comparison is not completely fair.
Unbinned methods show better performance. More information is included (event distributions, angular error estimate and the energy in the MLR method).
Among unbinned methods, the Expectation Maximization show better results (or at least equivalent) without using the expected performances of the detector. It is a reliable and robust method.
EM algorithm
Comparison among methods
Grid/Cluster
ML Ratio
EM algorithm
Probability to detect a source at 5σ and 3σ as a function of the observed average number of events from a source located at -80º per year.
50%: <8 events (<6 events) with 5σ (3σ).
Discovery power as a function of the mean number of observed events (after track reconstruction and quality cuts)

10. Discovery potential and sensitivity
The expected sensitivity of ANTARES in 365 days is of the same order that the present limits set by AMANDA (for the Northern Hemisphere), since the better angular resolution allows a better background rejection
Grid/Cluster
ML Ratio
EM algorithm
Sensitivity to a E-2 neutrino spectrum from a δ = - 60º declination point-like source for ANTARES, and NEMO (astro-ph/ ) and averaged over all declinations in the Northern Sky for IceCube (astro-ph/ v1) with a 90% C.L. as a function of the exposure of the detector

11. Conclusions
ANTARES has a good opportunity to detect point-like neutrino sources. Its good angular resolution, and its privileged location (Galactic Centre visible 63% of the time) make it a very promising neutrino telescope.
However theoretical models of cosmic neutrino production, and experimental data from other experiments like AMANDA, suggest that expected neutrinos fluxes are very low. Hence powerful searching algorithms have to be developed.
In ANTARES several searching algorithms were devised. Among them, unbinned methods show a better discovery potential and sensitivity. The Expectation-Maximization algorithm is a pattern recognition algorithm that can be applied to the search of point-like neutrino sources with very good results.
In one year, ANTARES is expected to reach the same sensitivity as AMANDA after 1001 days of duty cycle due to the better angular resolution of ANTARES.