Title: Estimating activity cycles with probabilistic methods I. Bayesian Generalised Lomb-Scargle Periodogram with Trend,
Abstract: Period estimation is one of the central topics in astronomical time series
analysis, where data is often unevenly sampled. Especially challenging are
studies of stellar magnetic cycles, as there the periods looked for are of the
order of the same length than the datasets themselves. The datasets often
contain trends, the origin of which is either a real long-term cycle or an
instrumental effect, but these effects cannot be reliably separated, while they
can lead to erroneous period determinations if not properly handled. In this
study we aim at developing a method that can handle the trends properly, and by
performing extensive set of testing, we show that this is the optimal procedure
when contrasted with methods that do not include the trend directly to the
model. The effect of the form of the noise (whether constant or
heteroscedastic) on the results is also investigated. We introduce a Bayesian
Generalised Lomb-Scargle Periodogram with Trend (BGLST), which is a
probabilistic linear regression model using Gaussian priors for the
coefficients and uniform prior for the frequency parameter. We show, using
synthetic data, that when there is no prior information on whether and to what
extent the true model of the data contains a linear trend, the introduced BGLST
method is preferable to the methods which either detrend the data or leave the
data untrended before fitting the periodic model. Whether to use noise with
different than constant variance in the model depends on the density of the
data sampling as well as on the true noise type of the process.