Detecting Self-similarity in a Time Series

Wavelet analysis is proving to be a very powerful tool for characterizing behavior, especially self-similar behavior, over a wide range of time scales.

In 1993, Scargle and colleagues at NASA-Ames Research Center and elsewhere investigated the quasiperiodic oscillations (QPOs) and very low-frequency noise (VLFN) from an astronomical X-ray accretion source, Sco X-1 as possibly being caused by the same physical phenomenon (12). Sco X-1 is part of a close binary star system in which one member is a late main sequence star and the other member (Sco X-1) is a compact star generating bright X rays. The causes for QPOs in X-ray sources have been actively investigated in the past, but other aperiodic phenomena such as VLFNs have not been similarly linked in the models. Their Sco X-1 data set was an interesting 5-20 keV EXOSAT satellite time-series consisting of a wide-range of time scales, from 2 ms to almost 10 hours.

Galactic X-ray sources are often caused by the accretion of gas from one star to another in a binary star system. The accreted object is usually a compact star such as a white dwarf, neutron star, or black hole. Gas from the less massive star flows to the other star via an accretion disk (that is, a disk of matter around the compact star flowing inward) around the compact star. The variable luminosities are caused by irregularities in the gas flow. The details of the gas flow are not well-known.

The researchers noticed that the luminosity of Sco X-1 varied in a self-similar manner, that is, the statistical character of the luminosities examined at different time resolutions remained the same. Since one of the great strengths of wavelets is that they can process information effectively at different scales, Scargle used a wavelet tool called a scalegram to investigate the time-series.

Scargle defines a scalegram of a time series as the average of the squares of the wavelet coefficients at a given scale. Plotted as a function of scale, it depicts much of the same information as does the Fourier power spectrum plotted as a function of frequency. Implementing the scalegram involves summing the product of the data with a wavelet function, while implementing the Fourier power spectrum involves summing the data with a sine or cosine function. The formulation of the scalegram makes it a more convenient tool than the Fourier transform because certain relationships between the different time scales become easier to see and correct, such as seeing and correcting for photon noise.

The scalegram for the time-series clearly showed the QPOs and the VLFNs, and the investigators were able to calculate a power-law to the frequencies. Subsequent simulations suggested that the cause of Sco-X1's luminosity fluctuations may be due to a chaotic accretion flow.