The EM algorithm applied to the search of high energy neutrino sources

The detection of astrophysical sources of high energy neutr inos is one of the most sought-after experimental goals in present astroparticle physics. The l ow number of signal events expected in high energy neutrino telescopes calls for powerful algorithms to d isentangle clusters of small number of events from the background. In this contribution we explore the pot entiality of the Expectation-Maximisation algorithm (EM) for the search of neutrino point-like sources i n a generic kilometre-scale neutrino telescope located in the Mediterranean sea. The EM algorithm, widely u sed in clustering analysis, has been adapted in this work to the special case of low signal statistics in a r el tively high background. We describe here how the problem has been tackled and compare our results to th e well-known binning technique. The method can also be applied to other similar problems like for instance the search of nearby ultra-high energy cosmic-ray sources. The Expectation-Maximisation algorithm The Expectation-Maximisation algorithm [1] is a general approach to maximum likelihood estimation for finite mixture model problems. In these models different groups in the data are described by different density components, so that the total probability density function (pdf) can be expressed asp(x) = ∑g j=1 πj p(x; θj), whereg is the number of mixture components, πj ≥ 0 are the mixing proportions that satisfy the unitary relation ( ∑g j=1 πj = 1) andp(x; θj), j = 1, . , g, are thecomponent density functions which depend on a parameter vector θj . The values of πj and the components ofθj have to be found by maximising the likelihood. The number and form of the component density functions depend on the problem being tackled. Frequently, p(x; θj) is taken to be the multivariate normal (Gaussian) density parametrised by a mean and a covariance matrix Σj , (θj = {μj ,Σj}). In this particular case the equations given by the EM algorithm are wellknown. Having a data sample of n observations, the likelihood for a mixture model withg density components is given byL(Ψ) = ∏n i=1 ∑g j=1 πjp(xi; θj) where Ψ stands for the set of parameters {π1, . , πg; θ1, . , θg}. In general it is not possible to solve explicitly∂L/∂Ψ = 0 and an iterative approach must be used. The EM algorithm is a general iterative procedure to maximise mixture model likelihoods. The idea is to assume that the set of observations given in the data make up a set ofincompletedata vectors{x}. The likelihood given by thisincompletedata set can be expressed by L(Ψ) = p({x},Ψ). The unknown information missed in the data sample is as a matter of fact whether an observation belongs to a component or to another, in other words, we lack the information about the clustering structure of the population. Let {y} denote acompletedata set, a version of{x} formed by a new vector yi = (xi, zi), wherezi = (zi1, ., zig) is simply a class indicator vector, that is,zik = 1, if xi belongs to groupk, andzik = 0, otherwise. With thiscompletedata set, the total density function is: g(y;Ψ) = g(x, z;Ψ) = p(x; z,Ψ)f(z;Ψ) (1) and the likelihood isL(Ψ) = g({y},Ψ). We can obtain back the likelihood of the previous incompletedata set fromg({y},Ψ) by integrating over Proceedings of the 30th International Cosmic Ray Conference Rogelio Caballero, Juan Carlos D’Olivo, Gustavo Medina-Tanco, Lukas Nellen, Federico A. Sánchez, José F. Valdés-Galicia (eds.) Universidad Nacional Autónoma de México, Mexico City, Mexico, 2008 Vol. 4 (HE part 1), pages 809–812 ID 640