Estimating activity cycles with probabilistic methods

Context. Debate over the existence of branches in the stellar activity-rotation diagrams continues. Application of modern time series analysis tools to study the mean cycle periods in chromospheric activity index is lacking. Aims. We develop such models, based on Gaussian processes (GPs), for one-dimensional time series and apply it to the extended Mount Wilson Ca H&K sample. Our main aim is to study how the previously commonly used assumption of strict harmonicity of the stellar cycles as well as handling of the linear trends affect the results. Methods. We introduce three methods of different complexity, starting with Bayesian harmonic regression model, followed by GP regression models with periodic and quasi-periodic covariance functions. We also incorporate a linear trend as one of the components. We construct rotation to magnetic cycle period ratio-activity (RCRA) diagrams and apply a Gaussian mixture model to learn the optimal number of clusters explaining the data. Results. We confirm the existence of two populations in the RCRA diagram; this finding is robust with all three methods used. We find only one significant trend in the inactive population, namely that the cycle periods get shorter with increasing rotation, leading to a positive slope in the RCRA diagram. This is in contrast with earlier studies, that postulate the existence of trends of different types in both of the populations. Our data is consistent with only two activity branches (inactive, transitional) instead of three (inactive, active, transitional) such that the active branch merges together with the transitional one. The retrieved stellar cycles are uniformly distributed over the $$\langle R_{{\rm{HK}}}^\prime \rangle $$ activity index, indicating that the operation of stellar large-scale dynamos carries smoothly over the Vaughan-Preston gap. At around the solar activity index, however, indications of a disruption in the cyclic dynamo action are seen. Conclusions. Our study shows that stellar cycle estimates from time series the length of which is short in comparison to the searched cycle itself depend significantly on the model applied. Such model-dependent aspects include the improper treatment of linear trends, while the assumption of strict harmonicity can result in the appearance of double cyclicities that seem more likely to be explained by the quasi-periodicity of the cycles. In the case of quasi-periodic GP models, which we regard the most physically motivated ones, only 15 stars were found with statistically significant cycles against red noise model. The periodicities found have to, therefore, be regarded as suggestive