On the Correct Estimate of the Probability of False Detection of the
  Matched Filter in Weak-Signal Detection Problems
  The detection reliability 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 optimising 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.
