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Signals with different local standard deviations

Figure 4: Scaling behavior of nonstationary correlated signals with different local standard deviation. (a) Anti-correlated signal ($\alpha =0.1$) with standard deviation $\sigma _1=1$ and amplified segments with standard deviation $\sigma _2=4$. The size of each segment is $W=20$ and the fraction of the amplified segments is $p=0.1$ from the total length of the signal ( $N_{max}=2^{20}$). (b) Scaling behavior of the signal in (a) for a different fraction $p$ of the amplified segments (after normalization of the globe standard deviation to unity). A crossover from anti-correlated behavior ($\alpha =0.1$) at small scales to random behavior ($\alpha =0.5$) at large scales is observed. (c) Dependence of the crossover scale $n_{\times }$ on the fraction $p$ of amplified segments for the signal in (a). $n_{\times }$ is determined from the difference $\Delta $ of $\log_{10}[F(n)/n]$ between the nonstationary signal with amplified segments and the original stationary signal. Here we choose $\Delta =0.04$. (d) Scaling behavior of nonstationary signals obtained from correlated stationary signals ($1>\alpha >0.5$) with standard deviation $\sigma _1=1$, for a different fraction of the amplified segments with $\sigma _2=4$. No difference in the scaling is observed, compared to the original stationary signal.
\begin{figure}\centerline{
\epsfysize=0.35\columnwidth{\epsfbox{mixamp.eps}}}\vs...
...terline{
\epsfysize=0.55\columnwidth{\epsfbox{dfa1_amp_a07n20.eps}}}\end{figure}

Here we consider nonstationary signals comprised of segments with the same local scaling exponent, but different local standard deviations. We first generate a stationary correlated signal $u(i)$ (see Sec. 2) with fixed standard deviation $\sigma _1=1$. Next, we divide the signal $u(i)$ into non-overlapping segments of size $W$. We then randomly choose a fraction $p$ of the segments and amplify the standard deviation of the signal in these segments, $\sigma _2=4$ [Fig.4(a)]. Finally, we normalize the entire signal to global standard deviation $\sigma =1$ by dividing the value of each point of the signal by $\sqrt{(1-p)\sigma_1^2+p\sigma_2^2}$.

For nonstationary anti-correlated signals ($\alpha <0.5$) with segments characterized by two different values of the standard deviation, we observe a crossover at scale $n_{\times }$ [Fig.4(b)]. For small scales $n<n_{\times }$, the behavior is anti-correlated with an exponent equal to the scaling exponent $\alpha $ of the original stationary anti-correlated signal $u(i)$. For large scales $n>n_{\times }$, we find a transition to random behavior with exponent $0.5$, indicating that the anti-correlations have been destroyed. The dependence of crossover scale $n_{\times }$ on the fraction $p$ of segments with larger standard deviation is shown in Fig. 4(c). The dependence is not monotonic because for $p=0$ and $p=1$, the local standard deviation is constant throughout the signal, i.e., the signal becomes stationary and thus there is no crossover. Note the asymmetry in the value of $n_{\times }$ -- a much smaller value of $n_{\times }$ for $p=0.05$ compared to $p=0.95$ [see Fig. 4(b-c)]. This result indicates that very few segments with a large standard deviation (compared to the rest of the signal) can have a strong effect on the anti-correlations in the signal. Surprisingly, the same fraction of segments with a small standard deviation (compared to the rest of the signal) does not affect the anti-correlations up to relatively large scales.

For nonstationary correlated signals ($\alpha >0.5$) with segments characterized by two different values of the standard deviation, we surprisingly find no difference in the scaling of $F(n)/n$, compared to the stationary correlated signals with constant standard deviation [Fig. 4(d)]. Moreover, this observation remains valid for different sizes of the segments $W$ and for different values of the fraction $p$ of segments with larger standard deviation. We note that in the limiting case of very large values of $\sigma_{2}/\sigma_{1}$, when the values of the signal in the segments with standard deviation $\sigma_{1}$ could be considered close to ``zero'', the results in Fig. 4(d) do not hold and we observe a scaling behavior similar to that of the signal in Fig. 5(c) (see following section).


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Next: Signals with different local Up: Signals with different local Previous: Signals with different local
Zhi Chen 2002-08-28