Pseudononstationarity in the scaling exponents of finite-interval time series

The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of $N$ observations is known for finite variance processes to vary as $\sim 1∕N$ as $N\to \infty$ for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with $N$ for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable $N$, one is far from this $\sim 1∕N$ limiting behavior. We propose a semiempirical estimate for the minimum $N$ needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series.

Figure 1

(Color online) (a) Time series of length $N={10}^6$ for the $p$-model $(p=0.6)$. (b) Variation of the second-order moment of the increments $y_i(t,\tau )$ for time scale $\tau =1$ of the above time series where the original time series has been partitioned into $L=100$ and 10 equal-sized intervals. (c) Variation of conditioned ${\zeta }_i\left(2\right)$ with time for the $p$-model with the same segmentation as in (b)—also shown are the mean values of the exponents for different partitioning corresponding to different sample sizes. (d) Same as (c) but with errors explicitly shown.

Figure 2

(Color online) (a) The variance of conditioned $\zeta \left(2\right)$ with sample size $N$ for all the synthetic finite variance processes studied shown on a log-log plot. (b) same as in (a) for all the synthetic infinite variance processes studied. The diagonal dashed line on both these plots indicates a negative slope of unit gradient so that comparison with theoretically expected asymptotic behavior can be made. The vertical dashed lines indicate the areas outside of which errors begin to dominate due to (i) (bottom vertical line) lack of values of ${\zeta }_i\left(2\right)$ to make a decent estimate of $\mathrm{var}\mathbf{\left(}\zeta \left(2\right)\mathbf{\right)}$, and (ii.) (top vertical line) failure of the iterative conditioning technique to obtain unbiased estimates of $\zeta \left(2\right)$.

Figure 3

(Color online) Plots showing how errors can be ascribed to the plots in Fig. 2. The top plots show the results of the study for 10 different randomly seeded realizations of sample size $N={10}^6$ for (a) a standard Brownian motion and (b) an $\alpha$-stable Lévy motion $(\alpha =1.4)$. Plots (c) and (d) are the mean averages of the realizations in (a) and (b), respectively, where the error bars are calculated from the maximum deviation from this mean in the 10 realizations.

Figure 4

(Color online) Plot of the variance of conditioned $\zeta \left(2\right)$ with sample size $N$ for all the real-world data sets studied shown on a log-log plot. The dashed line indicates a negative slope of unit gradient. Also included for comparison are the archetypal synthetic data sets for the finite variance and infinite variance processes.

Figure 5

(Color online) PDF’s $H\mathbf{\left(}\zeta \left(2\right)\mathbf{\right)}$ obtained for (a) a standard Brownian motion, (b) $\alpha$-stable Lévy process $(\alpha =1.4)$, (c) $p$-model $(p=0.6)$ and (d) ACE 2000 $B^2$ for two different sample sizes (i) $N=1000$ ($N=500$ for ACE) and (ii) $N=10000$ ($N=5000$ for ACE). The sample PDF’s are overlaid with maximum likelihood estimate (MLE) model fits of a normal distribution for (a), (c), and (d), and Gumbel max-stable (extreme value type I) fits for (b). For both these models, $\mu$ is the location parameter and $\sigma$ is the scale parameter, which for the normal distribution coincide with the mean and standard deviation. The samples of ${\zeta }_i\left(2\right)$ (varying with time $t$) from which these PDFs are constructed are shown in the corresponding insets of each plot.