Abstract
The minimal spanning tree (MST) algorithm is a graph-theoretical cluster-finding method. We previously applied it to γ-ray bidimensional images, showing that it is quite sensitive in finding faint sources. Possible sources are associated with the regions where the photon arrival directions clusterize. MST selects clusters starting from a particular “tree” connecting all the point of the image and performing a cut based on the angular distance between photons, with a number of events higher than a given threshold. In this paper, we show how a further filtering, based on some parameters linked to the cluster properties, can be applied to reduce spurious detections. We find that the most efficient parameter for this secondary selection is the magnitude M of a cluster, defined as the product of its number of events by its clustering degree. We test the sensitivity of the method by means of simulated and real Fermi-Large Area Telescope (LAT) fields. Our results show that \(\sqrt{M}\) is strongly correlated with other statistical significance parameters, derived from a wavelet based algorithm and maximum likelihood (ML) analysis, and that it can be used as a good estimator of statistical significance of MST detections. We apply the method to a 2-year LAT image at energies higher than 3 GeV, and we show the presence of new clusters, likely associated with BL Lac objects.
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Notes
The formal definition of a spatial Poisson process with uniform density involves the fact that in each closed subregion the number of points follow a Poisson distribution, and the various closed disjoint regions are independent from each other. See, e.g., Diggle (2003) for details.
As an example, for a field containing 50,000 photons, the computational time is shorter than 20 seconds on a 2.2 GHz Intel Core i5 laptop with 4 GB RAM. The secondary selection algorithms account for more than 90 % of the overall running time.
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Acknowledgements
We are grateful to the referee, Maria Concetta Maccarone, whose helpful comments greatly improved the quality of the text, and to Toby Burnett, Seth Digel and Andrea Tramacere for useful discussions. This research has been partially supported by a grant from Università di Roma “La Sapienza”. The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K.A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.
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Campana, R., Bernieri, E., Massaro, E. et al. Minimal spanning tree algorithm for γ-ray source detection in sparse photon images: cluster parameters and selection strategies. Astrophys Space Sci 347, 169–182 (2013). https://doi.org/10.1007/s10509-013-1488-0
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DOI: https://doi.org/10.1007/s10509-013-1488-0