The correct estimate of the probability of false detection of the matched filter in weak-signal detection problems

The reliable detection of weak signals is a critical issue in many astronomical contexts and may have severe consequences for determining number counts and luminosity functions, but also for optimizing the use of telescope time in follow-up observations. Because of its optimal properties, one of the most popular and widely-used detection technique is the matched filter (MF). This is a linear filter designed to maximise the detectability of a signal of known structure that is buried in additive Gaussian random noise. In this work we show that in the very common situation where the number and position of the searched 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, this technique, when applied in its standard form, may severely underestimate the probability of false detection. This is because the correct use of the MF relies upon a priori knowledge of the position of the signal of interest. In the absence of this information, the statistical significance of features that are actually noise is overestimated and detections claimed that are actually spurious. For this reason, we present an alternative method of computing the probability of false detection that is based on the probability density function (PDF) of the peaks of a random field. It is able to provide a correct estimate of the probability of false detection for the one-, two-and three-dimensional case. We apply this technique to a real two-dimensional interferometric map obtained with ALMA.