Noise

The standard signals we generate in our study are uncorrelated, correlated, and anticorrelated noise. First we must have a clear idea of the scaling behaviors of these standard signals before we use them to study the effects from other aspects. We generate noises by using a modified Fourier filtering method[63]. This method can efficiently generate noise, $u(i)$ ( $i=1,2,3,...,N_{\mbox{\scriptsize max}}$), with the desired power-law correlation function which asymptotically behaves as: $<\vert\sum\limits_{j=i}^{i+t}u(j)\vert^2> \sim t^{2\alpha}$. By default, a generated noise has standard deviation $\sigma=1$. Then we can test DFA and R/S by applying it on generated noises since we know the expected scaling exponent $\alpha$.

Figure 14: Scaling behavior of noise with the scaling exponent $\alpha$. The length of noise $N_{\mbox{\scriptsize max}}=2^{17}$. (a) Rescaled range analysis (R/S) (b) Order 1 detrended fluctuation analysis (DFA-1) (c) Order 2 detrended fluctuation analysis. We do the linear fitting for R/S analysis and DFA-1 in three regions as shown and get $\alpha _1$, $\alpha _2$ and $\alpha _3$ for estimated $\alpha$, which are listed in the Table IV and Table V. We find that the estimation of $\alpha$ is different in the different region.

Before doing that, we want to briefly review the algorithm of R/S analysis. For a signal $u(i)$( $i=1,...,N_{\mbox{\scriptsize max}}$), it is divided into boxes of equal size $n$. In each box, the cumulative departure, $X_i$ (for $k$-th box, $i=kn+1,..., kn+n$), is calculated

$$X_i =\sum\limits_{j=kn+1}^{i} (u(j)-<u>)$$ (25)

where $<u>=n^{-1}\sum\limits_{i=kn+1}^{(k+1)n}u(i)$ , and the rescaled range $R/S$ is defined by

$$R/S = S^{-1} \left[\max \limits_{kn+1\leq i \leq (k+1)n}X_i - \min\limits_{kn+1\leq i \leq (k+1)n}X_i\right],$$ (26)

where $S= \sqrt{{n^{-1}}\sum\limits_{j=1}^{n}(u(j)-<u>)^2}$ is the standard deviation in each box. The average of rescaled range in all the boxes of equal size $n$, is obtained and denoted by $<R/S>$. Repeat the above computation over different box size $n$ to provide a relationship between $<R/S>$ and $n$. According to Hurst's experimental study[64], a power-law relation between $<R/S>$ and the box size $n$ indicates the presence of scaling: $<R/S> \sim n^{\alpha}$.

Figure 14 shows the results of R/S, DFA-1 and DFA-2 on the same generated noises. Loosely speaking, we can see that $F(n)$ (for DFA) and $R/S$ (for R/S analysis) show power-law relation with $n$ as expected: $F(n) \sim n^{\alpha}$ and $R/S \sim n^{\alpha}$. In addition, there is no significant difference between the results of different order DFA except for some vertical shift of the curves and the little bend-down for small box size $n$. The bent-down for very small box of $F(n)$ from higher order DFA is because there are more variables to fit those few points.

Table IV: Estimated $\alpha$ of correlation noise from R/S analysis in three regions as shown in Fig.14(a). $\alpha$ is the input value of the scaling exponent, $\alpha _1$ is the estimated in the region 1 for $4<n\leq32$, $\alpha _2$ in the region 2 for $32<n\leq 3162$ and $\alpha _3$ in the region 3 for $3126<n \leq 2^{17}$. Noise are the same as used in Table V.

$\alpha$	$\alpha _1$	$\alpha _2$	$\alpha _3$
0.1	0.44	0.23	0.12
0.3	0.52	0.37	0.23
0.5	0.62	0.52	0.47
0.7	0.72	0.70	0.45
0.9	0.81	0.87	0.63

Table V: Estimated $\alpha$ of correlation noise from DFA-1 in three regions as shown in Fig.14(b). $\alpha$ is the input value of the scaling exponent, $\alpha1$ is the estimated in the region 1 for $4<n\leq32$, $\alpha2$ in the region 2 for $32<n\leq 3162$ and $\alpha3$ in the region 3 for $3126<n \leq 2^{17}$.

$\alpha$	$\alpha _1$	$\alpha _2$	$\alpha _3$
0.1	0.28	0.15	0.08
0.3	0.40	0.31	0.22
0.5	0.55	0.50	0.35
0.7	0.72	0.69	0.55
0.9	0.91	0.91	0.69

Ideally, when analyzing a standard noise, $F(n)$ (DFA) and $R/S$ ($R/S$ analysis) will be a power-law function with a given power: $\alpha$, no matter which region of $F(n)$ and $R/S$ is chosen for calculation. However, a careful study shows that the scaling exponent $\alpha$ depends on scale $n$. The estimated $\alpha$ is different for the different regions of $F(n)$ and $R/S$ as illustrated by Figs. 14(a) and 14(b) and by Tables IV and V. It is very important to know the best fitting region of DFA and R/S analysis in the study of real signals. Otherwise, the wrong $\alpha$ will be obtained if an inappropriate region is selected.

In order to find the best region, we first determine the dependence of the locally estimated $\alpha$, $\alpha_{\mbox{\scriptsize loc}}$, on the scale $n$. First, generate a standard noise with given scaling exponent $\alpha$; then calculate $F(n)$ (or $R/S$), and obtain $\alpha_{\mbox{\scriptsize loc}}(n)$ by local fitting of $F(n)$ (or $R/S$). Same random simulation is repeated 50 times for both DFA and R/S analysis. The resultant average $\alpha_{\mbox{\scriptsize loc}}(n)$, respectively, are illustrated in Fig.15 for DFA-1 and R/S analysis.

If a scaling analysis method is working properly, then the result $\alpha_{\mbox{\scriptsize loc}}(n)$ from simulation with $\alpha$ would be a horizontal line with slight fluctuation centered about $\alpha_{\mbox{\scriptsize loc}}(n) = \alpha$. Note from Fig.15 that such a horizontal behavior does not hold for all the scales $n$ but for a certain range from $n_{\mbox{\scriptsize min}}$ to $n_{\mbox{\scriptsize max}}$. In addition, at small scale, R/S analysis gives $\alpha_{\mbox{\scriptsize loc}} > \alpha$ if $\alpha < 0.7$ and $\alpha_{\mbox{\scriptsize loc}} < \alpha$ if $\alpha > 0.7$, which has been pointed out by Mandelbrot[65] while DFA gives $\alpha_{\mbox{\scriptsize loc}} > \alpha$ if $\alpha <1.0$ and $\alpha_{\mbox{\scriptsize loc}} < \alpha$ if $\alpha > 1.0$.

It is clear that the smaller the $n_{\mbox{\scriptsize min}}$ and the larger the $n_{\mbox{\scriptsize max}}$, the better the method. We also perceive that the expected horizontal behavior stops because the fluctuations become larger due to the under-sampling of $F(n)$ or $R/S$ when $n$ gets closer to the length of the signal $N_{\mbox{\scriptsize max}}$. Furthermore, it can be seen from Fig.15 that $n_{\mbox{\scriptsize max}} \approx \frac{1}{10}N_{\mbox{\scriptsize max}}$ independent of $\alpha$ (if the best fit region exists), which is why one tenth of the signal length is the maximum box size when using DFA or R/S analysis.

Figure 15: The estimated $\alpha$ from local fit (a) R/S analysis, the length of signal $N_{\mbox{\scriptsize max}}=2^{14}$. (b)R/S analysis, $N_{\mbox{\scriptsize max}}=2^{20}$. (c) DFA-1, $N_{\mbox{\scriptsize max}}=2^{14}$ (d) DFA-1, $N_{\mbox{\scriptsize max}}=2^{20}$. $\alpha_{\mbox{\scriptsize loc}}$ come from the average of $50$ simulations. If a technique is working, then the data for scaling exponent $\alpha$ should be a weakly fluctuating horizontal line centered about $\alpha_{\mbox{\scriptsize loc}} = \alpha$. Note that such a horizontal behavior does not hold for all the scales. Generally, such a expected behavior begins from some scale $n_{\mbox{\scriptsize min}}$, holds for a range and ends at a larger scale $n_{\mbox{\scriptsize max}}$. For DFA-1, $n_{\mbox{\scriptsize min}}$ is quite small $\alpha > 0.5$. For R/S analysis, $n_{\mbox{\scriptsize min}}$ is small only when $\alpha \approx 0.7$.

On the contrary, $n_{\mbox{\scriptsize min}}$ does not depend on the $N_{\mbox{\scriptsize max}}$ since $\alpha_{\mbox{\scriptsize loc}}(n)$ at small $n$ hardly changes as $N_{\mbox{\scriptsize max}}$ varies but it does depend on $\alpha$. Thus, we obtain $n_{\mbox{\scriptsize min}}$ quantitatively as shown in Fig.16. For R/S analysis, only for $\alpha \approx 0.7$, $n_{\mbox{\scriptsize min}}$ is small; for $\alpha$ a little away from $0.7$ (for example, 0.5), $n_{\mbox{\scriptsize min}}$ becomes very large and close to $n_{\mbox{\scriptsize max}}$, indicating that the best fit region will vanish and R/S analysis does not work at all. Comparing to R/S, DFA works better since $n_{\mbox{\scriptsize min}}$ is quite small for $\alpha > 0.5$ correlated signals.

Figure 16: The starting point of good fit region, $n_{\mbox{\scriptsize min}}$, for DFA-1 and R/S analysis. The results are obtained from 50 simulations, in which the length of noise is $N_{\mbox{\scriptsize max}}=2^{20}$. The condition for a good fit is $\Delta \alpha = \vert\alpha_{\mbox{\scriptsize loc}} - \alpha\vert< 0.01$. The data for $\alpha > 1.0$ shown in the shading area are obtained by applying analysis on the integrations of noises with $\alpha <1.0$. It is clear that DFA-1 works better than R/S analysis because its $n_{\mbox{\scriptsize min}}$ is always smaller than that of R/S analysis.

One problem remains for DFA, $n_{\mbox{\scriptsize min}}$ for small $\alpha$ ($\leq 0.5$) is still too large comparing to those for large $\alpha$($>0.5$). We can improve it by applying DFA on the integration of the noise with $\alpha < 0.5$. The resultant new expected $\alpha^{'}$ for the integrated signal would be $\alpha^{'}_0 = \alpha+1$, while the $n_{\mbox{\scriptsize min}}$ for the integrated signal becomes much smaller as shown also in Fig.16(shading area $\alpha > 1$). Therefore, for a noise with $\alpha < 0.5$, it is best to estimate the scaling exponent $\alpha^{'}$ of the integrated signal first and then obtain $\alpha$ by $\alpha = \alpha^{'}-1$. This is what we did in the following sections to those anticorrelated signals.