From periodic to cyclic processes in stellar magnetic activity research: time series analysis methods and their applications

One of the unanswered questions in stellar activity research is how the rotation period and the magneticcycle period of a star are related. A prerequisite to answering this question is being able to estimate both of these quantities as reliably as possible. Throughout the years, the prevailing methods have mostly been based on the well-known Lomb-Scargle periodogram. However, such a periodogram and its analogues are hard to interpret, when the input signal is not fully periodic. Observations of the solar cycle properties through factors, such as, the sunspot number over time, and non-linear dynamo models both clearly indicate that the stellar dynamo process is indeed quasi-periodic and non-stationary. Hence, a more correct approach is to relax the assumption of periodicity. The development and application of such methods is the main aim of this thesis. To investigate stellar cycles theoretically, the most advanced approach is to use global 3D magnetoconvection models solving the full MHD equations. These have only recently started to show similar quasi-periodic behaviour as the observed datasets. Real and simulated data pose completely different requirements for the analysis methods. While the former are unevenly sampled and sparse, the latter contain vast amounts of multidimensional data. For the estimation of magnetic cycles, an additional problem with observational data is their relative shortness. Throughout the thesis I will thoroughly address the above aspects. In this work, several methods have been developed for analysing time series of active stars. Carrier fit (CF)method is a simple and efficient way for fitting a continuous model into the time series of active stars. Side by side with this method a visualisation technique is used, which allows deviations from strict periodicity at different times to be easily detected, revealing the quasi-periodic and non-stationary effects. Another method, called D2 phase dispersion statistic is a robust tool for estimating periods of a quasi-periodic time series. It allows a simple generalisation to multiple dimensions, which is useful when analysing datasets of 3D magnetoconvection simulations. We also use probabilistic models for period estimation. For short datasets, the period estimates can become sensitive to the ways the linear trend in the data is handled. We show that for proper treatment one needs to include the trend component in the model, while using prior distributions for regularisation. Other probabilistic models, which have been used in the study include Gaussian processes (GPs) with periodic and quasi-periodic covariance functions. From the toolbox of methods suitable for non-stationary data, we have used ensemble empirical mode decomposition (EEMD). Our applications involve a young solar analogue LQ Hya, 3D magnetoconvection simulation called PENCILMillennium and a Mount Wilson (MW) stellar chromospheric activity dataset. For LQ Hya, we estimated the mean rotation period, surface differential rotation coefficient and fitted a continuous light curve model using the CF method. In the case of PENCIL-Millennium simulation data, we used both EEMD and the D2 statistic to extract the different dynamo modes with their locations in the convection zone. These modes include a five-year cycle, which is an analogue of the 22-year magnetic cycle of the Sun, and two much longer cycles. Furthermore, with the help of the D2 statistic, we were able to find a very incoherent short cycle with a period around half a year, which resembles the quasi-biennial oscillations of the Sun. In the analysis of the MW dataset, the main aim was to repeat the cycle length estimation with a simple harmonic model while properly handling trends, but also trying out periodic and quasi-periodic GP models. All three methods led to quite similar results, however, the reliability of the quasi-periodic model remained questionable due to the shortness of the datasets.We confirmed the existence of two different star populations in the activity diagram. However, as opposed to the formerly known positive correlations within both of these branches, we confirmed only a positive correlation within the inactive branch. The results were also compared to the recent 3D magnetoconvection simulations