Comparing significance criteria for cyclic modulations in time series

• Published in: Nuclear instruments & methods in physics research. Section A, Accelerators, spectrometers, detectors and associated equipment Accelerators, spectrometers, detectors and associated equipment, 2020-07, Vol.968, p.163933, Article 163933
• Main Author: Pommé, S.
• Format: Article
• Language: English
• Subjects: Cyclic modulation; Least-squares method; Linear regression; Lomb–Scargle periodogram; Statistical significance; Time series analysis
• ISSN: 0168-9002; 1872-9576
• DOI: 10.1016/j.nima.2020.163933

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. •Derived uncertainty propagation formulas for sinusoidal cycles in time series.•Solutions for weighted and unweighted fit, with and without a baseline shift.•Significance criterion for normalised amplitude based on Rayleigh distribution.•Significance test compared to power test with Lomb–Scargle periodogram.