A Bayesian test for periodic signals in red noise

Abstract

Many astrophysical sources, especially compact accreting sources, show strong, random brightness fluctuations with broad power spectra in addition to periodic or quasi-periodic oscillations (QPOs) that have narrower spectra. The random nature of the dominant source of variance greatly complicates the process of searching for possible weak periodic signals. We have addressed this problem using the tools of Bayesian statistics; in particular using Markov chain Monte Carlo techniques to approximate the posterior distribution of model parameters, and posterior predictive model checking to assess model fits and search for periodogram outliers that may represent periodic signals. The methods developed are applied to two example datasets, both long XMM-Newton observations of highly variable Seyfert 1 galaxies: RE J1034+396 and Mrk 766. In both cases a bend (or break) in the power spectrum is evident. In the case of RE J1034+396 the previously reported QPO is found but with somewhat weaker statistical significance than reported in previous analyses. The difference is due partly to the improved continuum modelling, better treatment of nuisance parameters, and partly to different data selection methods.Comment: 15 pages, 6 figures. Accepted for publication in MNRA

Bayesian timing analysis of giant flare of SGR 180620 by RXTE PCA

“…We used a procedure that does not require binning and considers the rotational modulation and decaying tail of the flare (Bretthorst 1988;Gregory & Loredo 1992, 1996Jaynes & Bretthorst 2003;Vaughan 2010).…”

Gamma-ray and optical oscillations of 0716+714, MRK 421, and BL Lacertae

“…Both families of models have been considered in the literature (e.g. Kelly et al 2009;Edelson et al 2013) and, within the limited frequency range of our analysis, both provide acceptable fits to the obtained PDS basing on Kolmogorov-Smirnov (KS) tests of the residuals against a χ 2 distribution (Vaughan 2010) with two degrees of freedom. A discussion of specific physical meanings for the adopted noise models is beyond the scope of this paper yet, as a general rule, PLs tend to predict higher noise at the lowest frequencies than AR1 models, although PLs often show a more uniform distribution of residuals.…”

“…Finally, we also draw the T R = max j R j statistics to evaluate the global significance of any peak (i.e. the probability that, at any frequency, the power is equal or larger than a 8 http://dan.iel.fm/emcee/current/ chosen value ) in the PDS (see Vaughan 2010, for a deeper discussion), where R = 2P/S , P the simulated or observed PDS, and S the best-fit PDS model. Given that the same procedure is applied to the simulated, as well as to the real data, there is no need to perform a multiple trial correction owing to the (typically unknown) number of independent sampled frequencies.…”

“…Essentially, we followed the procedure described in Vaughan (2005Vaughan ( , 2010 and Guidorzi et al (2016). Power density spectra (PDS) were derived by means of generalized Lomb-Scargle periodograms (e.g.…”

X-ray timing analysis of the quasar PG 1211+143

“…For a given model, we found the best-fitting model parameters by minimizing the Whittle likelihood, S, as discussed in Vaughan (2010) and Barret & Vaughan (2012). We fitted the broadband 0.2-10 keV PSD with two models: (i) a simple power law, and (ii) a bending power law (see González-Martín & Vaughan 2012 for discussion of the models and fitting methods).…”