Signals with segments removed

In this section, we study the effect of nonstationarity caused by removing segments of a given length from a signal and stitching together the remaining parts -- a ``cutting'' procedure often used in pre-processing data prior to analysis. To address this question, we first generate a stationary correlated signal $u(i)$ (see Sec. 2) of length $N_{max}$ and a scaling exponent $\alpha$, using the modified Fourier filtering method[63]. Next, we divide this signal into $N_{max}/W$ non-overlapping segments of size $W$ and randomly remove some of these segments. Finally, we stitch together the remaining segments in the signal $u(i)$ [Fig. 2(a)], thus obtaining a surrogate nonstationary signal which is characterized by three parameters: the scaling exponent $\alpha$, the segment size $W$ and the fraction of the signal $u(i)$, which is removed.

Figure 2: Effects of the ``cutting'' procedure on the scaling behavior of stationary correlated signals. $N_{max}=2^{20}$ is the number of points in the signals (standard deviation $\sigma =1$) and $W$ is the size of the cutout segments. (a) A stationary signal with 10% of the points removed. The removed parts are presented by shaded segments of size $W=20$ and the remaining parts are stitched together. (b) Scaling behavior of nonstationary signals obtained from an anti-correlated stationary signal (scaling exponent $\alpha <0.5$) after the cutting procedure. A crossover from anti-correlated to uncorrelated ($\alpha =0.5$) behavior appears at scale $n_{\times }$. The crossover scale $n_{\times }$ decreases with increasing the fraction of points removed from the signal. We determine $n_{\times }$ based on the difference $\Delta$ between the logarithm of $F(n)/n$ for the original stationary anti-correlated signal ($\alpha =0.1$) and the nonstationary signal with cutout segments: $n_{\times }$ is the scale at which $\Delta \geq 0.04$. Dependence of the crossover scale $n_{\times }$ on the fraction (c) and on the size $W$ (d) of the cutout segments for anti-correlated signals with different scaling exponent $\alpha$. (e) Cutting procedure applied to correlated signals ($\alpha >0.5$). In contrast to (b), no discernible effect on the scaling behavior is observed for different values of the scaling exponent $\alpha$, even when up to 50% of the points in the signals are removed.

We find that the scaling behavior of such a nonstationary signal strongly depends on the scaling exponent $\alpha$ of the original stationary correlated signal $u(i)$. As illustrated in Fig. 2(b), for a stationary anti-correlated signal with $\alpha =0.1$, the cutting procedure causes a crossover in the scaling behavior of the resultant nonstationary signal. This crossover appears even when only $1\%$ of the segments are cut out. At the scales larger than the crossover scale $n_{\times }$ the r.m.s. fluctuation function behaves as $F(n) \sim n^{0.5}$, which means an uncorrelated randomness, i.e., the anti-correlation has been completely destroyed in this regime. For all anti-correlated signals with exponent $\alpha <0.5$, we observe a similar crossover behavior. This result is surprising, since researchers often take for granted that a cutting procedure before analysis does not change the scaling properties of the original signal. Our simulation shows that this assumption is not true, at least for anti-correlated signals.

Next, we investigate how the two parameters -- the segment size $W$ and the fraction of points cut out from the signal -- control the effect of the cutting procedure on the scaling behavior of anti-correlated signals. For the fixed size of the segments ($W=20$), we find that the crossover scale $n_{\times }$ decreases with increasing the fraction of the cutout segments [Fig. 2(c)]. Furthermore, for anti-correlated signals with small values of the scaling exponent $\alpha$, e.g., $\alpha =0.1$ and $\alpha =0.2$, we find that $n_{\times }$ and the fraction of the cutout segments display an approximate power-law relationship. For a fixed fraction of the removed segments, we find that the crossover scale $n_{\times }$ increases with increasing the segment size $W$ [Fig. 2(d)]. To minimize the effect of the cutting procedure on the correlation properties, it is advantageous to cut smaller number of segments of larger size $W$. Moreover, if the segments which need to be removed are too close (e.g., at a distance shorter than the size of the segments), it may be advantageous to cut out both the segments and a part of the signal between them. This will effectively increase the size of the segment $W$ without substantially changing the fraction of the signal which is cut out, leading to an increase in the crossover scale $n_{\times }$. Such strategy would minimize the effect of this type of nonstationarity on the scaling properties of data. For small values of the scaling exponent $\alpha$ ($\alpha<0.25$), we find that $n_{\times }$ and $W$ follow power-law relationships [Fig. 2(d)]. The reason we do not observe a power-law relationship between $n_{\times }$ and $W$ and between $n_{\times }$ and the fraction of cutout segments for the values of the scaling exponent $\alpha$ close to $0.5$ may be due to the fact that the crossover regime becomes broader when it separates scaling regions with similar exponents, thus leading to uncertainty in defining $n_{\times }$. For a fixed $W$ and a fixed fraction of the removed segments [see Figs. 2(c) and (d)], we observe that $n_{\times }$ increases with the increasing value of the scaling exponent $\alpha$, i.e., the effect of the cutting procedure on the scaling behavior decreases when the anti-correlations in the signal are weaker ($\alpha$ closer to $0.5$).

Finally, we consider the case of correlated signals $u(i)$ with $1.5>\alpha>0.5$. Surprisingly, we find that the scaling of correlated signals is not affected by the cutting procedure. This observation remains true independently of the segment size $W$ -- from very small $W=5$ up to very large $W=5000$ segments -- even when up to $50\%$ of the segments are removed from a signal with $N_{max}\sim 10^{6}$ points [Fig. 2(e)].