Issue 
A&A
Volume 627, July 2019



Article Number  A103  
Number of page(s)  9  
Section  Numerical methods and codes  
DOI  https://doi.org/10.1051/00046361/201834854  
Published online  08 July 2019 
The correct estimate of the probability of false detection of the matched filter in weaksignal detection problems
III. Peak distribution method versus the Gumbel distribution method
^{1}
Chip Computers Consulting s.r.l., Viale Don L. Sturzo 82, S.Liberale di Marcon, 30020 Venice, Italy
email: robertovio@tin.it
^{2}
ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany
email: pandrean@eso.org
^{3}
Department of Physics Graduate School of Science, The University of Tokyo, 731 Hongo, Bunkyo, Tokyo 1130033, Japan
Received:
13
December
2018
Accepted:
7
June
2019
The matched filter (MF) represents one of the main tools to detect signals from known sources embedded in the noise. In the Gaussian isotropic case, the noise can be assumed to be the realization of a Gaussian random field (GRF). The most important property of the MF, the maximization of the probability of detection subject to a constant probability of false detection or false alarm (PFA), makes it one of the most popular techniques. However, the MF technique relies upon the a priori knowledge of the number and the position of the searched signals in the GRF (e.g. an emission line in a spectrum or a pointsource on a map), which usually are not available. A typical way out is to assume that, if present, the position of a signal coincides with one of the peaks in the matched filtered data. A detection is claimed when the probability that a given peak is due only to the noise (i.e. the PFA) is smaller than a prefixed threshold. This last step represents a critical point in the detection procedure. Since a signal is searched for amongst the peaks, the probability density function (PDF) of the amplitudes of the latter has to be used for the computation of the PFA. Such a PDF, however, is different from the Gaussian. Moreover, the probability that a detection is false depends on the number of peaks present in the filtered GRF. This is because the greater the number of peaks in a GRF, the higher the probability of peaks due to the noise that exceed the detection threshold. If this fact is not taken into account, the PFA can be severely underestimated. In statistics this is a wellknown problem named the multiple comparisons, multiple testing, or multiple hypotheses problem, whereas in other fields it is known as the lookelsewhere effect. Many solutions have been proposed to this problem. However, most of them are of a nonparametric type hence not able to exploit all the available information. Recently, this limitation has been overcome by means of two efficient parametric approaches. One is explicitly based on the PDF of the peak amplitudes of a smooth and isotropic GRF whereas the other makes use of the Gumbel distribution, which represents the asymptotic PDF of the corresponding extreme. On the basis of numerical experiments as well of an application to an interferometric map obtained with the Atacama Large Millimeter/submillimeter Array (ALMA), we show that, although the two methods produce almost identical results, the first is more flexible and at the same time allows us to check the reliability of the detection procedure.
Key words: methods: data analysis / methods: statistical
© ESO 2019
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