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Register a new userOn the Correct Estimate of the Probability of False Detection of the Matched Filter in Weak-Signal Detection Problems
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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.
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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.
The matched filter (MF) is one of the most popular and reliable techniques to the detect signals of known structure and amplitude smaller than the level of the contaminating noise. Under the assumption of stationary Gaussian noise, MF maximizes the probability of detection subject to a constant probability of false detection or false alarm (PFA). This property relies upon a priori knowledge of the position of the searched signals, which is usually not available. Recently, it has been shown that when applied in its standard form, MF may severely underestimate the PFA. As a consequence the statistical significance of features that belong to noise is overestimated and the resulting detections are actually spurious. For this reason, an alternative method of computing the PFA has been proposed that is based on the probability density function (PDF) of the peaks of an isotropic Gaussian random field. In this paper we further develop this method. In particular, we discuss the statistical meaning of the PFA and show that, although useful as a preliminary step in a detection procedure, it is not able to quantify the actual reliability of a specific detection. For this reason, a new quantity is introduced called the specific probability of false alarm (SPFA), which is able to carry out this computation. We show how this method works in targeted simulations and apply it to a few interferometric maps taken with the Atacama Large Millimeter/submillimeter Array (ALMA) and the Australia Telescope Compact Array (ATCA). We select a few potential new point sources and assign an accurate detection reliability to these sources.
We present a conceptual design study of external calibrators in the 21 cm experiment towards detecting the globally averaged radiation of the epoch of reionization (EoR). Employment of external calibrator instead of internal calibrator commonly used in current EoR experiments allows to remove instrumental effects such as beam pattern, receiver gain and instability of the system if the conventional three-position switch measurements are implemented in a short time interval. Furthermore, in the new design the antenna system is placed in an underground anechoic chamber with an open/closing ceiling to maximally reduce the environmental effect such as RFI and ground radiation/reflection. It appears that three of the four external calibrators proposed in this paper, including two indoor artificial transmitters and one outdoor celestial radiation (the Galactic polarization), fail to meet our purpose. Diurnal motion of the Galactic diffuse emission turns to be the most possible source as an external calibrator, for which we have discussed the observational strategy and the algorithm of extracting the EoR signal.
Detection of templates (e.g., sources) embedded in low-number count Poisson noise is a common problem in astrophysics. Examples include source detection in X-ray images, gamma-rays, UV, neutrinos, and search for clusters of galaxies and stellar streams. However, the solutions in the X-ray-related literature are sub-optimal -- in some cases by considerable factors. Using the lemma of Neyman-Pearson we derive the optimal statistics for template detection in the presence of Poisson noise. We demonstrate that this method provides higher completeness, for a fixed false-alarm probability value, compared with filtering the image with the point-spread function (PSF). In turn, we find that filtering by the PSF is better than filtering the image using the Mexican-hat wavelet (used by wavedetect). For some background levels, our method improves the sensitivity of source detection by more than a factor of two over the popular Mexican-hat wavelet filtering. This filtering technique can also be used also for fast PSF photometry and flare detection, and it is efficient, as well as straight forward to implement. We provide an implementation in MATLAB.
Current and future interferometeric gravitational-wave detectors are limited predominantly by shot noise at high frequencies. Shot noise is reduced by introducing arm cavities and signal recycling, however, there exists a tradeoff between the peak sensitivity and bandwidth. This comes from the accumulated phase of signal sidebands when propagating inside the arm cavities. One idea is to cancel such a phase by introducing an unstable optomechanical filter. The original design proposed in [Phys.~Rev.~Lett.~{bf 115},~211104 (2015)] requires an additional optomechanical filter coupled externally to the main interferometer. Here we consider a simplified design that converts the signal-recycling cavity itself into the unstable filter by using one mirror as a high-frequency mechanical oscillator and introducing an additional pump laser. However, the enhancement in bandwidth of this new design is less than the original design given the same set of optical parameters. The peak sensitivity improvement factor depends on the arm length, the signal-recycling cavity length, and the final detector bandwidth. For a 4~km interferometer, if the final detector bandwidth is around 2~kHz, with a 20~m signal-recycling cavity, the shot noise can be reduced by 10 decibels, in addition to the improvement introduced by squeezed light injection. We also find that the thermal noise of the mechanical oscillator is enhanced at low frequencies relative to the vacuum noise, while having a flat spectrum at high frequencies.
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