On the significance of modulations in time series

Abstract

Analytical equations are derived for the uncertainty of the amplitude of cyclic modulations obtained by a least-squares fit of a sinusoidal function to a time series. A general equation is presented for a weighted fit to unevenly spaced data. A simple solution is derived for randomly distributed data with constant relative uncertainty and compared to a formerly published heuristic relationship. Different scenarios are considered in which, besides the amplitude, also the phase shift and/or an offset value are free parameters. A free fit of the phase shift parameter transforms the normal probability distribution of the amplitude into a Rayleigh distribution. The additional fit of an offset value increases the width due to the change in degrees of freedom. A time series analysis is applied to the residuals of an exponential fit to experimental decay curves of various nuclides, in search of cyclic deviations to the exponential decay law. The statistical significance of the modulations is discussed.