On the use of structure functions to study blazar variability: caveats and problems

• Published in: Monthly notices of the Royal Astronomical Society Vol. 404; no. 2; pp. 931 - 946
• Main Authors: Emmanoulopoulos, D., McHardy, I. M., Uttley, P.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 11.05.2010; Wiley-Blackwell; Oxford University Press
• Subjects: Astronomy; Astrophysics; BL Lacertae objects: general; Earth, ocean, space; Exact sciences and technology; galaxies: active; galaxies: individual: Mrk 501; methods: data analysis; methods: numerical; methods: statistical; Quasars; Stars & galaxies; Statistical analysis; Statistical methods; methods: statistical; methods: data analysis; BL Lacertae objects: general; galaxies: individual: Mrk 501; methods: numerical; galaxies: active; Data analysis; Uncertainty; Estimator; Variability; Light curves; Power spectral density; Structure functions; Numerical method; Blazar; BL Lacertae objects; Active galaxies; Statistical method; Statistical models; Cosmic radio sources; Shot noise
• ISSN: 0035-8711; 1365-2966
• DOI: 10.1111/j.1365-2966.2010.16328.x

The extensive use of the structure function (SF) in the field of blazar variability across the electromagnetic spectrum suggests that characteristics time-scales are embedded in the light curves of these objects. We argue that for blazar variability studies, the SF results are sometimes erroneously interpreted leading to misconceptions about the actual source properties. Based on extensive simulations, we caution that spurious breaks will appear in the SFs of almost all light curves, even though these light curves may contain no intrinsic characteristic time-scales, i.e. having a featureless underlying power spectral density (PSD). We show that the time-scales of the spurious SF breaks depend mainly on the length of the artificial data set and also on the character of the variability i.e. the shape of the PSD. The SF is often invoked in the framework of shot-noise models to determine the temporal properties of individual shots. We caution that although the SF may be fitted to infer the shot parameters, the resultant shot-noise model is usually inconsistent with the observed PSD. As any model should fit the data in both the time and the frequency domain, the shot-noise model, in these particular cases, cannot be valid. Moreover, we show that the lack of statistical independence between adjacent SF points, in the standard SF formulation, means that it is not possible to perform robust statistical model fitting following the commonly used least-squares fitting methodology. The latter yields uncertainties in the fitting parameters (i.e. slopes, breaks) that are far too small with respect to their true statistical scatter. Finally, it is also commonly thought that SFs are immune to the sampling problems, such as data gaps, which affects the estimators of the PSDs. However, we show that SFs are also troubled by gaps which can induce artefacts.