Abstract
We demonstrate a simple method for testing the significance of peaks in the periodogram of red noise data. The procedure was designed to test for spurious periodicities in X-ray light curves of active galaxies, but can be used quite generally to test for periodic components against a background noise spectrum assumed to have a power law shape. The method provides a simple and fast test of the significance of candidate periodic signals in short, well-sampled time series such as those obtained from XMM-Newton observations of Seyfert galaxies, without the need for Monte Carlo simulations. A full account is made of the number of trials and the uncertainties inherent to the model fitting. Ignoring these subtle effects can lead to substantially overestimated significances. These difficulties motivate us to demand high standards of detection (minimum per cent confidence) for periodicities in sources that normally show red noise spectra. The method also provides a simple means to estimate the power spectral index, which may be an interesting parameter itself, regardless of the presence/absence of periodicities.
Highlights
Many astrophysical sources show erratic, aperiodic brightness fluctuations with steep power spectra
The logarithm of the periodogram ordinate, with the bias removed, is an unbiased estimator of the logarithm of the spectrum, and is distributed independently and identically at each frequency. (The raw periodogram points are not identically distributed in linear space since the scatter depends on the spectrum, which is a function of frequency.) we can use a Least squares (LS) fitting procedure to get a reasonable estimate of the power spectral slope α and normalisation N by fitting a linear function y = mx + c to the plot of log[I( f j)] versus log[ f j]
Not be used on non-uniformly sampled time series
Summary
Many astrophysical sources show erratic, aperiodic brightness fluctuations with steep power spectra. The power spectrum of these variations, which describes the dependence of the variability amplitude on temporal frequency, is often reasonably approximated as a simple power law (over at least a decade of frequency). XRBs often show quasi-periodic oscillations (QPOs) that show-up as peaks in the power spectrum over the continuum noise spectrum. 6.1.4 of Priestley (1981), and discussed in an astrophysical context by Leahy et al (1983) and van der Klis (1989) Without modification these methods cannot be used to test against red noise variations. S. Vaughan: A simple test for periodic signals in red noise only strictly valid when the underlying continuum spectrum is a power law. The appendix discusses a more generally applicable method of periodogram fitting (that makes no assumption on the form of the underlying spectrum)
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