Time Series Analysis of Unequally Spaced Data:
Intercomparison between the Schuster Periodogram and the LS-spectra

V.V.Vityazev
SPbU

Session ID: T1.03   Type: oral

Abstract:

It is generally believed that the classical Schuster periodogram must not be used in the problem of finding the periodicities in the irregularly spaced data. This follows from the fact that if the data is pure noise, the statistical distribution of the Schuster periodogram is no longer exponential. From this point of view, the so-called LS-spectra based on the least squares fitting of a sine function to the data are recommended. Nevertheless, in many situations the Schuster periodograms and the LS-spectra are close to being identical. The paper presents a comparative study of the Schuster periodogram and the LS-spectra with respect to their statistical properties. It is shown that the likeness of the periodograms under consideration depends on the properties of the spectral window corresponding to the distribution of time points. The main results are: a) all the estimators evaluated at frequency are identical if ; b) the Schuster periodogram differs from the LS-spectra at the frequency , where is the frequency at which the spectral window has a large side peak due to irregular distribution of time points; c) the analytical expression for the probability distribution of the Schuster periodogram when the time series is assumed to be unevenly spaced pure noise is found. It is shown that the probability distribution deviates from the exponential law only at the frequencies that satisfy the condition . The paterns of the time points yielding such pathology are given. The numerical examples for several situations typical in astronomy illustrate these conclusions.

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