Self-affinity of time series with finite domain power-law power spectrum

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Abstract
We consider time series with finite domain (band-limited) power-law spectra. Our analysis is based on exact representation of the structure (mean square increment) function for such processes and shows that the considered time series exhibit an approximate self-affinity in a wide range of time scales. The self-affinity is embodied in the leading asymptotic term which represents the familiar ‘‘pure’’ fractal behavior. Next we explicitly show that the impact of the lowest and the highest scales of the process cannot in general be neglected and conclude from this that for adequate description of the natural processes considerations beyond the simple fractal analysis are required. We also propose a method for determining the spectral parameters of experimentally recorded self-affine time series which is based on nonlinear, least-squares fit and the exact form of the structure function. Tests employing numerically generated series as a benchmark demonstrate this method’s excellent accuracy and robustness.