Self-affine time series: measures of weak and strong persistence

In this paper, we examine self-affine time series and their persistence. Time series are defined to be self-affine if their power-spectral density scales as a power of their frequency. Persistence can be classified in terms of range, short or long range, and in terms of strength, weak or strong. Self-affine time series are scale-invariant, thus they always exhibit long-range persistence. Synthetic self-affine time series are generated using the Fourier power-spectral method. We generate fractional Gaussian noises (fGns), −1β1, where β is the power-spectral exponent. These are summed to give fractional Brownian motions (fBms), 1β3, where the series are self-affine fractals with fractal dimension 1D2; β=2 is a Brownian motion. With β>1, the time series are non-stationary and moments of the time series depend upon its length; with β<1 the time series are stationary. We define self-affine time series with β>1 to have strong persistence and with β<1 to have weak persistence. We use a variety of techniques to quantify the strength of persistence of synthetic self-affine time series with −3β5. These techniques are effective in the following ranges: (1) semivariograms, 1β3, (2) rescaled-range (R/S) analyses, −1β1, (3) Fourier spectral techniques, all values of β, and (4) wavelet variance analyses, all values of β. Wavelet variance analyses lack many of the inherent problems that are found in Fourier power-spectral analysis.