K-Contact Distance for Noisy Nonhomogeneous Spatial Point Data with application to Repeating Fast Radio Burst sources

TL;DR

The paper tackles the challenge of inferring second-order characteristics from noisy, nonhomogeneous Poisson point data, with a focus on distinguishing physically independent sources from repeaters in CHIME/FRB FRB observations. It develops a hierarchical Bayesian model for a parameterized NHPP intensity that accounts for measurement noise and uses the posterior to derive predictive $k$-contact probabilities and their bounds. A novel analytic bound on the probability of $k$-contact coincidences is derived and validated against simulations, enabling efficient inference of the probability of coincidence $P_C$. Applied to CHIME/FRB, the method yields substantial improvements in repeater identification, with many candidates reclassified as unambiguous repeaters and a notable increase in detection significance, while also providing uncertainty quantification through posterior sampling. The approach is generalizable to other sparse, noisy spatial datasets and offers practical implications for FRB source modeling and cross-disciplinary hotspot detection.

Abstract

This paper introduces an approach to analyze nonhomogeneous Poisson processes (NHPP) observed with noise, focusing on previously unstudied second-order characteristics of the noisy process. Utilizing a hierarchical Bayesian model with noisy data, we estimate hyperparameters governing a physically motivated NHPP intensity. Simulation studies demonstrate the reliability of this methodology in accurately estimating hyperparameters. Leveraging the posterior distribution, we then infer the probability of detecting a certain number of events within a given radius, the $k$-contact distance. We demonstrate our methodology with an application to observations of fast radio bursts (FRBs) detected by the Canadian Hydrogen Intensity Mapping Experiment's FRB Project (CHIME/FRB). This approach allows us to identify repeating FRB sources by bounding or directly simulating the probability of observing $k$ physically independent sources within some radius in the detection domain, or the $\textit{probability of coincidence}$ ($P_{\text{C}}$). The new methodology improves the repeater detection $P_{\text{C}}$ in 91% of cases when applied to the largest sample of previously classified observations, with a median improvement factor (existing metric over $P_{\text{C}}$ from our methodology) of $\sim$ 4800.

Figures (8)

• Figure 1: $k$-contact probability bound versus frequency from simulations of two different NHPP intensity models and in different noise regimes. In each panel, 500 test points are drawn uniformly from the domain of the experiment, and then 50,000 simulated datasets are drawn from the relevant NHPP $\Lambda$ and positions are perturbed by Gaussian noise centered at $(0,0)$ and with standard deviation of $10^{-1}, 10^{-2},10^{-3}, 10^{-3}$ for the first, second, third, and fourth columns respectively. We record the frequency of having $k=2$ events within some radius $r=10^{-2}$ for each simulation. We then, given $s_0$, the noise distribution, and $r$, compute the probability bound in Equation \ref{['eqn:bound']} for the first three columns. The final column shows the estimated probability from Equation \ref{['eqn:berman']}, reproduced from kthnearest, ignoring the underlying positional uncertainty in the computation. Top row: Simulations for test centers ($s_0$) drawn from the bivariate Gaussian intensity (see Section \ref{['sec:boundsims']}). Bottom row: As for the top row, but simulated datasets are drawn from a bivariate Gaussian mixture intensity. A black line is drawn in each panel to denote where the bound and simulation frequency would be equal. Points are colored according to the value of the intensity at the test center, $\Lambda(s_0)$.
• Figure 2: Illustration of right ascension ($\alpha$) and declination ($\delta$). These two angles are equivalent to longitude and latitude on the celestial sphere, respectively. $\alpha$ is referenced from the 'vernal equinox', which is one of the intersecting points of the celestial equator and the geographical equator, and $\alpha$ is defined here between $0$ and $360$ degrees. $\delta$ differs from the typical spherical coordinate polar angle as it is referenced to the $x-y$ plane, or the plane of the celestial equator, rather than the $+z$-axis (north celestial pole). $\delta$ is defined between $-90$ (south celestial pole) and $90$ degrees (north celestial pole).
• Figure 3: Illustration depicting the horizon-to-horizon view of the CHIME telescope to show the zenith angle ($\zeta$) of a given source (indicated by a star) as a function of the declination of that source ($\delta$). Zenith is defined as perpendicular to the horizon. The north celestial pole and celestial equator are defined as perpendicular to one another, and the north celestial pole is the axis around which the Earth spins. The angle between the north celestial pole and the horizon is equal to the geographic latitude ($\phi$) of the telescope 2022ApJS..261...29C. The declination is defined as the angle between the source and the celestial equator. Hence we can calculate the zenith angle, $\zeta(\delta)$, of a source as $\zeta = \phi - \delta$.
• Figure 4: top panel: Sensitivity as a function of declination and normalized to attain a max value at 1 at our zenith, intended to compare the behaviour of our best sensitivity estimate and $\cos^b(\zeta)$ (black line). The sensitivity estimate was modeled using CHIME/FRB's publicly available beam-model, available at https://chime-frb-open-data.github.io/beam-model/ by summing the two components of CHIME/FRB's beam, the 'formed' beam and the 'primary' beam (a detailed description is more technical than appropriate to be included here, but can be found in overview). The sensitivity is expected to follow $\cos^b(\zeta)$ where $b$ is a constant which depends on how CHIME/FRB forms beams on the sky as well as the polarization of FRBs 2015AA...576A..62Noverview. We use a simple least squares fit in order to estimate a value for $b$ to show that the median of the sensitivity is well described by such a function. Bottom panel: residual values.
• Figure 5: An example of uncertainty regions for FRB localizations with CHIME/FRB's real-time pipeline, to demonstrate the multi-modal nature. The colormap shows which confidence interval percentage one would need to consider in order to include that position in the localization error. In this plot, the unit for $\alpha$ is not fractional degrees but rather hours and minutes of arc, where 360 degrees are equal to 24 archours, and one degree is 60 arcminutes. The dots in the upper left corner illustrate the pattern of CHIME/FRB's beams in which the burst was detected. Credit: 2021ApJS..257...59C.