Conclusions

In this paper we studied the effects of three different types of nonstationarities using the DFA correlation analysis method. Specifically, we consider sequences formed in three ways: (i) stitching together segments of signals obtained from discontinuous experimental recordings, or removing some noisy and unreliable segments from continuous recordings and stitching together the remaining parts; (ii) adding random outliers or spikes to a signal with known correlations, and (iii) generating a signal composed of segments with different properties -- e.g. different standard deviations or different correlations. We compare the difference between the scaling results obtained for stationary correlated signals and for correlated signals with artificially imposed nonstationarities.

(i) We find that removing segments from a signal and stitching together the remaining parts does not affect the scaling behavior of positively correlated signals ( $1.5 \geq \alpha > 0.5$), even when up to 50% of the points in these signals are removed. However, such a cutting procedure strongly affects anti-correlated signals, leading to a crossover from an anti-correlated regime (at small scales) to an uncorrelated regime (at large scales). The crossover scale $n_{\times }$ increases with increasing value of the scaling exponent $\alpha$ for the original stationary anti-correlated signal. It also depends both on the segment size and the fraction of the points cut out from the signal: (1) $n_{\times }$ decreases with increasing fraction of cutout segments, and (2) $n_{\times }$ increases with increasing segment size. Based on our findings, we propose an approach to minimize the effect of cutting procedure on the correlation properties of a signal: When two segments which need to be removed are on distances shorter than the size of the segment, it is advantageous to cut out both the segments and the part of the signal between them.

(ii) Signals with superposed random spikes. We find that for an anti-correlated signal with superposed spikes at small scales, the scaling behavior is close to that of the stationary anti-correlated signal without spikes. At large scales, there is a crossover to random behavior. For a correlated signal with spikes, we find a different crossover from uncorrelated behavior at small scales to correlated behavior at large scales with an exponent close to the exponent of the original stationary signal. We also find that the spikes with identical density and amplitude may cause strong effect on the scaling of an anti-correlated signal while they have a much smaller or no effect on the scaling of a correlated signal -- when the two signals have the same standard deviations. We investigate the characteristics of the scaling of the spikes only and find that the scaling behavior of the signal with random spikes is a superposition of the scaling of the signal and the scaling of the spikes. We analytically prove this superposition relation by introducing a superposition rule.

(iii) Signals composed of segments with different local properties. We find that
(a) For nonstationary correlated signals comprised of segments which are characterized by two different values of the standard deviation, there is no difference in the scaling behavior compared to stationary correlated signals with constant standard deviation. For nonstationary anti-correlated signals, we find a crossover at scale $n_{\times }$. For small scales $n<n_{\times }$, the scaling behavior is similar to that of the stationary anti-correlated signals with constant standard deviation. For large scales $n>n_{\times }$, there is a transition to random behavior. We also find that very few segments with large standard deviation can strongly affect the anti-correlations in the signal. In contrast, the same fraction of segments with standard deviation smaller than the standard deviation of the rest of the anti-correlated signal has much weaker effect on the scaling behavior -- $n_{\times }$ is shifted to larger scales.
(b) For nonstationary signals consisting of segments with different correlations, the scaling behavior is a superposition of the scaling of the different components -- where each component contains only the segments exhibiting identical correlations and the remaining segments are replaced by zero. Based on our findings, we propose an approach to identify the composition of such complex signals: A first step is to ``guess'' the type of correlations from the DFA results at small and large scales. A second step is to determine the parameters defining the components, such as the segment size and the fraction of non-zero segments. We studied how the scaling characteristics of the components depend on these parameters and provide analytic scaling expressions.