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The detection reliability of weak signals is a critical issue in manyastronomical contexts and may have severe consequences for determining numbercounts and luminosity functions, but also for optimising the use of telescopetime in follow-up observations. Because of its optimal properties, one of themost popular and widely-used detection technique is the matched filter (MF).This is a linear filter designed to maximise the detectability of a signal ofknown structure that is buried in additive Gaussian random noise. In this workwe show that in the very common situation where the number and position of thesearched signals within a data sequence (e.g. an emission line in a spectrum)or an image (e.g. a point-source in an interferometric map) are unknown, thistechnique, when applied in its standard form, may severely underestimate theprobability of false detection. This is because the correct use of the MFrelies upon a-priori knowledge of the position of the signal of interest. Inthe absence of this information, the statistical significance of features thatare actually noise is overestimated and detections claimed that are actuallyspurious. For this reason, we present an alternative method of computing theprobability of false detection that is based on the probability densityfunction (PDF) of the peaks of a random field. It is able to provide a correctestimate of the probability of false detection for the one-, two- andthree-dimensional case. We apply this technique to a real two-dimensionalinterferometric map obtained with ALMA.

On the Correct Estimate of the Probability of False Detection of the Matched Filter in Weak-Signal Detection Problems