A simpler model with sinusoids only will be used when analyzing time series of activity indices. That is,
5







6







7





where 





 is some measurement of an activity index (e.g., emission in the 





 line, S-index, etc.) obtained with instrument INS. The parentheses in 





 mean that fits with and without long-term trends will also be investigated. The significance assessments here are performed using frequentist estimations of the false alarm probability (FAP) following the recipes by Baluev (2009). Detecting a periodic signal in a time series will consist of  comparing a null model (e.g., no 





 term) with a model with one more periodic signal (e.g., one sin/cos term). Since the period is a highly nonlinear parameter, many periods (about 20,000 equally spaced in 1/P or frequency domain) will be tested between 2 and 50,000 days. A plot of the improvement of the logarithm of the maximum likelihood statistic (





) against the period of the test signal produces a plot that is often called a periodogram. Periodograms for unevenly sampled data were developed in the late 1970s (e.g., Lomb 1976; Scargle 1982). In those cases, the merit statistic plotted against the test period was the improvement in 





 or the F-ratio statistic. These early methods only accounted for a model with a single sin/cos component. A further refinement was later implemented by also including an offset (as our γ) as a free parameter to acknowledge the fact that the mean of the series was not known a priori. These periodograms were called floating mean periodograms (Ferraz-Mello 1981) or generalized least-squares (GLS, Zechmeister & Kürster 2009) periodograms. RRM15 relies on the use of GLS periodograms. GLS is adequate to identify and quantify the significance of periodicities if the data contain a single sinusoidal signal and uncertainties in the time series are realistic. Otherwise, analytic significance estimates of the favored solutions using GLS can only be considered to be approximate. In this sense, GLS is a very efficient method to guess the most likely signals left in residual data, but it cannot be used to provide a reliable assessment of its significance. This is discussed in the context of the basic model comparison in Anglada-Escudé & Tuomi (2015). Further refinements of the model (e.g., simultaneous fit to a trend, combining different instruments with different offsets, include basic models for unaccounted noise) require the use of more sophisticated tools beyond GLS, like the in-likelihood periodograms presented here. Note that the 





 statistic used by GLS and previous periodograms is a particular case of the ln-likelihood statistic with 





, 





, and only one instrument INS. Finding the maximum likelihood statistic requires adjusting all of the free parameters of the model at each test period. For these reasons, it produces much more robust significance estimates than its predecessors, especially when the description of the data requires models more complex than a single sinusoid. Reviewing the properties of the likelihood function in the context of periodic signal detection is not the topic of this article. We refer the interested reader to Baluev (2009, 2013) for more detailed discussions and formal proofs.
Citation: Lomb 1976
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