Signals with random spikes

In this section, we consider nonstationarity related to the presence of random spikes in data and we study the effect of this type of nonstationarity on the scaling properties of correlated signals. First, we generate surrogate nonstationary signals by adding random spikes to a stationary correlated signal $u(i)$ [see Sec. 2 and Fig. 3(a-c)].

Figure 3: Effects of random spikes on the scaling behavior of stationary correlated signals. (a) An example of an anti-correlated signal $u(i)$ with scaling exponent $\alpha =0.2$, $N_{max}=2^{20}$ and standard deviation $\sigma =1$. (b) A series of uncorrelated spikes ( $\alpha _{sp}=0.5$) at 5$\%$ randomly chosen positions (concentration $p=0.05$) and with uniformly distributed amplitudes $A_{sp}$ in the interval $[-4, 4]$. (c) The superposition of the signals in (a) and (b). (d) Scaling behavior of an anti-correlated signal $u(i)$ ($\alpha =0.2$) with spikes ($A_{sp}=1$, $p=0.05$, $\alpha _{sp}=0.5$). For $n<n_{\times }$, $F(n)/n \approx F_{\eta }(n)/n \sim n^{\alpha }$, where $F_{\eta }(n)/n$ is the scaling function of the signal $u(i)$. For $n>n_{\times }$, $F(n)/n \approx F_{sp}(n)/n \sim n^{\alpha _{sp}}$. (e) Scaling behavior of a correlated signal $u(i)$ ($\alpha =0.8$) with spikes ($A_{sp}=10$, $p=0.05$, $\alpha _{sp}=0.5$). For $n<n_{\times }$, $F(n)/n \approx F_{sp}(n)/n \sim n^{\alpha _{sp}}$. For $n>n_{\times }$, $F(n)/n \approx F_{\eta }(n)/n \sim n^{\alpha }$. Note that when $\alpha =\alpha _{sp}=0.5$, there is no crossover.

We find that the correlation properties of the nonstationary signal with spikes depend on the scaling exponent $\alpha$ of the stationary signal and the scaling exponent $\alpha_{sp}$ of the spikes. When uncorrelated spikes ( $\alpha _{sp}=0.5$) are added to a correlated or anti-correlated stationary signal [Fig 3(d) and (e)], we observe a change in the scaling behavior with a crossover at a characteristic scale $n_{\times }$. For anti-correlated signals ($\alpha <0.5$) with random spikes, we find that at scales smaller than $n_{\times }$, the scaling behavior is close to the one observed for the stationary anti-correlated signal without spikes, while for scales larger than $n_{\times }$, there is a crossover to random behavior. In the case of correlated signals ($\alpha >0.5$) with random 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 correlated signal. Moreover, we find that spikes with a very small amplitude can cause strong crossovers in the case of anti-correlated signals, while for correlated signals, identical concentrations of spikes with a much larger amplitude do not affect the scaling. Based on these findings, we conclude that uncorrelated spikes with a sufficiently large amplitude can affect the DFA results at large scales for signals with $\alpha <0.5$ and at small scales for signals with $\alpha >0.5$.

To better understand the origin of this crossover behavior, we first study the scaling of the spikes only [see Fig. 3(b)]. By varying the concentration $p$ ($0\leq p\leq 1$) and the amplitude $A_{sp}$ of the spikes in the signal, we find that for the general case when the spikes may be correlated, the r.m.s. fluctuation function behaves as

$$F_{sp}(n)/n=k_0\sqrt{p}A_{sp}n^{\alpha_{sp}},$$ (4)

where $k_0$ is a constant and $\alpha_{sp}$ is the scaling exponent of the spikes.

Next, we investigate the analytical relation between the DFA results obtained from the original correlated signal, the spikes and the superposition of signal and spikes. Since the original signal and the spikes are not correlated, we can use a superposition rule (see [61] and Appendix 7.1) to derive the r.m.s. fluctuation function $F(n)/n$ for the correlated signal with spikes:

$$[F(n)/n]^2=[F_{\eta}(n)/n]^2+[F_{sp}(n)/n]^2,$$ (5)

where $F_{\eta }(n)/n$ and $F_{sp}(n)/n$ are the r.m.s. fluctuation function for the signal and the spikes, respectively. To confirm this theoretical result, we calculate $\sqrt{[F_{\eta}(n)/n]^2+[F_{sp}(n)/n]^2}$ [see Figs. 3(d), (e)] and find this Eq. (5) is remarkably consistent with our experimental observations.

Using the superposition rule, we can also theoretically predict the crossover scale $n_{\times }$ as the intercept between $F_{\eta }(n)/n$ and $F_{sp}(n)/n$, i.e., where $F_{\eta}(n_{\times})=F_{sp}(n_{\times})$. We find that

$$n_{\times} = \left(\sqrt{p}A_{sp}\frac{k_0}{b_0}\right)^{1/(\alpha-\alpha_{sp})},$$ (6)

since the r.m.s. fluctuation function for the signal and the spikes are $F_{\eta}(n)/n=b_0 n^{\alpha}$ [61] and $F_{sp}(n)/n=k_0\sqrt{p}A_{sp}n^{\alpha_{sp}}$ [Eq. (4)], respectively. This result predicts the position of the crossover depending on the parameters defining the signal and the spikes.

Our result derived from the superposition rule can be useful to distinguish two cases: (i) the correlated stationary signal and the spikes are independent (e.g., the case when a correlated signal results from the intrinsic dynamics of the system while the spikes are due to external perturbations); and (ii) the correlated stationary signal and the spikes are dependent (e.g., both the signal and the spikes arise from the intrinsic dynamics of the system). In the latter case, the identity in the superposition rule is not correct (see Appendix 7.1).