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

TL;DR

This paper highlights that the standard matched filter technique can underestimate false detection probabilities in weak-signal detection when signal positions are unknown, and proposes a new method based on the PDF of random field peaks for accurate estimation.

Contribution

It introduces an alternative approach to accurately estimate false detection probabilities in weak-signal detection scenarios where signal positions are unknown, improving reliability over standard methods.

Findings

Standard matched filter underestimates false detection probability when signal positions are unknown.

The proposed method accurately estimates false detection probabilities in 1D, 2D, and 3D cases.

Application to ALMA data demonstrates the method's effectiveness in real interferometric maps.

Abstract

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.