# Correct estimate of the probability of false detection of the matched filter in weak-signal detection problems. III (Peak distribution method versus the Gumbel distribution method)

## TL;DR

This paper compares peak distribution and Gumbel distribution methods for accurately estimating false detection probabilities in weak-signal detection using matched filters, highlighting the advantages of the peak distribution approach.

## Contribution

It introduces and evaluates two parametric methods for false alarm probability estimation, demonstrating the flexibility and reliability of the peak distribution method over the Gumbel distribution.

## Key findings

- Both methods produce similar results in simulations.
- The peak distribution method offers greater flexibility.
- Simulations and ALMA maps validate the approaches.

## Abstract

The matched filter (MF) represents one of the main tools to detect signals from known sources embedded in the noise. In the Gaussian case the noise is 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 which usually are not available. A typical way out is to assume that 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. In this case the probability density function (PDF) of the amplitudes has to be used for the computation of the PFA, which is different from the Gaussian. Moreover, the probability that a detection is false depends on the number of peaks present in the filtered GRF, 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 not taken into account, the PFA can be severely underestimated. Many solutions proposed to this problem are non-parametric hence not able to exploit all the available information. This limitation has been overcome by means of two efficient parametric approaches, one based on the PDF of the peak amplitudes of a smooth and isotropic GRF whereas the other uses the Gumbel distribution (the asymptotic PDF of the corresponding extreme). Simulations and ALMA maps show that, although the two methods produce almost identical results, the first is more flexible and allows us to check the reliability of the detection procedure.