Testing whether a signal is strictly periodic

Abstract

Searches for periodicity in time series are often done with models of periodic signals, whose statistical significance is assessed via false alarm probabilities or Bayes factors. However, a statistically significant periodic model might not originate from a strictly periodic source. In astronomy in particular, one expects transient signals that show periodicity for a certain amount of time before vanishing. This situation is encountered, for instance, in the search for planets in radial velocity data. While planetary signals are expected to have a stable phase, amplitude, and frequency – except when strong planet-planet interactions are present – signals induced by stellar activity will typically not exhibit the same stability. In the present article we explore the use of periodic functions multiplied by time windows to diagnose whether an apparently periodic signal is truly so. We suggest diagnostics to check whether a signal is consistently present in the time series and has a stable phase, amplitude, and period. The tests are expressed both in a periodogram and Bayesian framework. Our methods are applied to the solar HARPS-N data as well as HD 215152, HD 69830, and HD 13808. We find that the HARPS-N solar data exhibit signals at the solar rotation period and its first harmonic (~13.4 days). The frequency and phase of the 13.4 day signal appear constant within the estimation uncertainties, but its amplitude presents significant variations that can be mapped to activity levels. Secondly, as previously reported, we find four, three, and two planets orbiting HD 215152, HD 69830, and HD 13808, respectively.

Detecting exoplanets with the false inclusion probability

“…Mortier & Collier Cameron (2017) suggests computing the periodogram adding one point at a time to check whether the evolution of a peak amplitude is compatible with a purely sinusoidal origin. Gregory (2016) and Hara et al (2022b) suggest to include in the model a so-called apodisation term, which is a multiplicative factor of the Keplerian signals that determines whether they are consistent through time or transient. Zucker (2015Zucker ( , 2016 suggests using a Hoeffding test, based on the phase-folded data.…”

A quarter century of spectroscopic monitoring of the nearby M dwarf Gl 514. A super-Earth on an eccentric orbit moving in and out of the habitable zone

“…The 1 method searches for several signals simultaneously and accounts for correlated noise at the stellar rotation period. As a result, signals that could appear dominant To further examine their properties, we apply the method of Hara et al (2022), which aims at testing whether the signals have consistent amplitude, phase and frequencies over time. We compute the apodized sine periodogram (ASP), defined as follows.…”

“…with the hypothesis that model K(ω, t (τ,ω) , τ, A , B ) is correct, where the fitted cosine and sine amplitudes A , B are obtained by fitting model K to the data. D z can easily be expressed as a generalised χ 2 distribution, with mean and variance given by an analytical expression (Hara et al 2022). In Fig.…”

Semi-supervised standardized detection of extrasolar planets

“…Because the sampling grid is irregular, an RV detection is most often performed through the Lomb-Scargle periodogram (LSP; Scargle 1982) or variants of the periodograms (see, e.g., Ferraz-Mello 1981, Cumming et al 1999, Cumming 2004, Reegen 2007, Bourguignon et al 2007, Zechmeister & Kürster 2009, Baluev 2013a, Gregory 2016, Hara et al 2017, Hara et al 2022. While the marginal distribution of the periodogram components at each frequency is often known for white Gaussian noise (WGN) of known variance (e.g., a χ 2 distribution for the LSP), the joint distribution of the periodogram components is much more difficult to characterize because the components exhibit dependences that are dictated by the sampling grid (or equivalently, by the spectral window).…”