An Exploration of Three New Methods to Facilitate the Calculation of the Significance Levels of Periodogram Peaks by Simulation

Abstract

Abstract At a given frequency ν, calculation of the periodogram entails evaluating two linear combinations and of the observations. It is pointed out that C and S are normally distributed even for relatively small numbers of observations. This can be used as a basis for simulating periodograms from a multivariate normal distribution with specified covariance structure. The advantage is that computationally expensive evaluations of sinusoids need to be performed only once—in the calculation of the covariances. Furthermore, the large covariance matrix can be factorized, and the factors replaced by low-rank approximations, which alleviates the computer memory demands. The second part of the paper discusses the choice of the degree of oversampling of the periodogram, i.e., the spacing of the frequency grid over which it is calculated. It is shown using , where ΔT is the time baseline of the observations, leads to an error of only about 1% in the percentiles of the distribution of maximum peak values. The third part of the paper deals with the representation of the distribution of periodogram maxima by the generalized extreme value distribution. Although the generalized extreme value form may formally differ highly significantly from the actual distribution of periodogram maxima, it is demonstrated that it may nonetheless be useful in practical terms. Results are illustrated throughout using three data sets with widely different aliasing properties.