Comparing significance criteria for cyclic modulations in time series

General solutions are derived for least-squares fits of amplitude, phase shift and baseline shift of sinusoidal modulations to normally distributed time series. Four cases are compared: weighted and unweighted fit, with and without a baseline shift. Computer simulations have been performed of the statistical distribution of the amplitude resulting from fits to white noise. A normalised amplitude is defined which follows exactly a Rayleigh distribution, even for small, uneven-sampled data sets. Equations are provided for the width of the distribution and the cumulative probability function. A simple criterion for the statistical significance of the amplitude fitted to a time series is derived and compared for equivalence with the Lomb–Scargle criterion based on a power function.