Extracting the scaling exponents of a self-affine, non-Gaussian process from a finite-length time series

We address the generic problem of extracting the scaling exponents of a stationary, self-affine process realized by a time series of finite length, where information about the process is not known a priori. Estimating the scaling exponents relies upon estimating the moments, or more typically structure functions, of the probability density of the differenced time series. If the probability density is heavy tailed, outliers strongly influence the scaling behavior of the moments. From an operational point of view, we wish to recover the scaling exponents of the underlying process by excluding a minimal population of these outliers. We test these ideas on a synthetically generated symmetric $\alpha$-stable Lévy process and show that the Lévy exponent is recovered in up to the 6th order moment after only $\sim 0.1–0.5\%$ of the data are excluded. The scaling properties of the excluded outliers can then be tested to provide additional information about the system.

Figure 1

(Color online) Plots showing probability density functions of the Lévy distribution for index $\alpha =1.4$ $(N={10}^6)$ at different values of differenced interval $\tau$ (a) before and (b) after the scaling collapse described by Eq. (15).

Figure 2

(Color online) Plots of (a) generalized structure functions $S^p$ vs $\tau$ for moments of order $p=1$–6, and (b) the scaling exponents $\zeta \left(p\right)$ vs $p$ (solid black line). These quantities are shown for a Lévy process of index $\alpha =1.4$ and with $N={10}^6$ data points. The dashed line indicates the expected scaling $\zeta \left(p\right)=p∕\alpha$ for $p<\alpha$; the dotted-dashed line indicates the scaling exponent observed for $p>\alpha$ in a finite-sized sample. The vertical arrow at $p\simeq \alpha$ separates these two regions of scaling.

Figure 3

(Color online) Plot showing the PDF, in Eq. (30), of the $m$th largest value of a sample size $N$ of a set of measurements taken from a Lévy-like process; the Lévy index $\alpha =1.5$.

Figure 4

(Color online) Plots showing the exponents $\zeta \left(p\right)$ against moment order $p$ of the generalized structure functions for various values of the percentage of large events excluded for (a) $\alpha =1.0$ and (b) $\alpha =1.8$. The arrows indicate the percentage beyond which convergence to the expected behavior $\zeta \left(p\right)=p∕\alpha$ is established. Both plots are for a sample size of $N={10}^6$.

Figure 5

(Color online) Plots showing the rapid convergence of the Lévy parameter $\alpha$; and the exponents of the 2nd and 3rd moments $\zeta \left(2\right)$ and $\zeta \left(3\right)$. The plots in (a) are for $\alpha =1.0$ and in (b) for $\alpha =1.8$—both have $N={10}^6$. $\zeta \left(2\right)$ and $\zeta \left(3\right)$ are the best fit gradients of the $S^p$ vs $\tau$ plots, and $\alpha$ is obtained from the inverse of the gradient of the $\zeta \left(p\right)$ vs $p$ plot shown in Fig. 4.

Figure 6

(Color online) $\zeta \left(p\right)$ vs $p$ plots for $\alpha =1.0$; (a) $N={10}^5$ and (b) $N=5\times {10}^6$. The arrows indicate the percentage beyond which convergence to the expected behavior $\zeta \left(p\right)=p∕\alpha$ is established.

Figure 7

(Color online) Log-log plot illustrating the scaling of the $m$th largest event $y_m^{*}$ with $\tau$ as $m$ is increased; $\alpha =1.8$, $N={10}^6$. For comparison with previous figures we indicate the percent of points that would be excluded for the particular $m$.

Figure 8

(Color online) Log-log plot illustrating the scaling of the largest event $y_m^{*}$ with $m∕N$ for various values of $\tau$; $\alpha =1.8$, $N={10}^6$.