Accuracy of period determination

Abstract

Periods of oscillation are frequently found using one of two methods: least-squares (LSQ) fit or power spectrum. Their errors are estimated using the LSQ correlation matrix or the Rayleigh resolution criterion δ νR=1/Δ T, respectively. In this paper we demonstrate that both estimates are statistically incorrect. On the one hand the LSQ covariance matrix does not account for correlation of residuals from the fit. Neglect of the correlations may cause large underestimation of the variance. On the other hand the Rayleigh resolution criterion is insensitive to signal-to-noise ratio and thus does not reflect quality of observations. We derive the correct variance estimates for the two methods. In the process we demonstrate that centre of the power spectrum line is a maximum likelihood estimate of frequency of the oscillation and demonstrate it is statistically equivalent to fitting of a sinusoid by LSQ, so the methods are statistically equivalent. Our new and correct variance estimate is quite simple and practicable. It is using the autocorrelation function (ACF) of the residuals to determine their mean correlation length and is valid under certain assumptions. We tested the extent to which the assumptions may be relaxed by numerical simulations.