# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 2 Theoretical Background
### Long-Range Correlations Measures (DFA and Hurst)
In short, \(\alpha\approx 0.5\) indicates that the signal is uncorrelated or non-similar in variability structure, as in the case of white noise (Figure 3d). If the signal's variability structures are long-range correlated, persistent, or self-similar, i.e., if a small-amplitude signal sample is very likely to be followed by an even smaller one (or similarly, if a large-amplitude sample tends to be followed by an even larger one), one finds \(\alpha>0.5\). That is the case with pink noise. Finally, if \(\alpha<0.5\), signal correlation is anti-persistent (the blue noise in Figure 3d): a small-amplitude sample is more likely followed by a large-amplitude one, and vice-versa. Figure 3: Basic parameters involved in standard \(R/\text{S}^{*}\) analysis: (**a**) a non-stationary time series \(x(i)\); (**b**) Example of Hurst exponent considering \(n=250\), where \(\langle x\rangle\) is the mean and \(\text{S}(n)\) the standard deviation, \(X(i,n)\) are the local differences accumulated from the mean, and \(R(n)\) is the local range, that is, the difference between maximum and minimum \(X(i,n)\) (adapted from [42]); (**c**) log-log plot of the rescaled range \(R/\text{S}^{*}\) versus window size \(n\). The estimated scaling exponent \(H\) is the slope of the least-squares fitting line; (**d**) representative examples of \(H\) value for a white (random), pink (fractal, long-range correlated, slightly persistent), and blue (anti- persistent) time series. The so-called \(R/\text{S}^{*}\) analysis is also a classic approach to evaluate the persistence or self-similarity of a signal. This methodology was introduced by [PERSON] [22], an English hydrologist, in an extensive series of empirical studies involving the flow of the Nile River. Its basic idea is to determine how the range of the cumulative fluctuations depend on the length of the data subset under analysis [42], being illustrated in Figure 3 for the same time series that was discussed in Figure 2a, considering windows with size \(n=250\)[43]. As in the DFA method, a time series of length \(N_{\text{max}}\) is divided into equal-length windows with size \(n\). Next, for each window:  
1. Take the mean \(\left\langle x\right\rangle_{n}=(1/n)\sum_{j=1}^{n}x(j)\) of the \(n\)-th window, marked as the solid red lines in Figure 3b. 2. Sum the differences from the mean to get the cumulative total \(X(i,n)\) at each data point, from the beginning of the period to any desired point, i.e., \(X(i,n)=\sum_{i=1}^{n}(x(i)-\left\langle x\right\rangle_{n})\), Figure 3b. 3. Calculate the local range \(R(n)=\max X(i,n)-\min X(i,n)\) as the maximum fluctuation of the sum of the deviation from the mean, where \(\max X(i,n)\) and \(\min X(i,n)\) are the maximum and minimum values of \(X(i,n)\), respectively, and \(1\leq i\leq n\) (Figure 3b). 4. Take the standard deviation \(\text{S}^{*}(n)=\sqrt{(1/n)\sum_{i=1}^{n}\left[x(i)-\left\langle x\right\rangle _{n}\right]^{2}}\) over the window to normalize the range relative to the input fluctuations in the series (dashed red lines in Figure 3b). 5. Rescale the range, that is, calculate \(R(n)/\text{S}^{*}(n)\). 6. Finally, calculate the mean value of the rescaled range for all windows, \(n_{uv}\) (four in the case of Figure 3b): \[\left(R/\text{S}^{*}\right)_{n}=\frac{1}{n_{uv}}\sum_{m=1}^{n_{uv}}R_{m}/ \text{S}^{*}{}_{m}\] (2)  
For windows with different lengths, the new dimensionless value \(R/\text{S}^{*}\) is well described by the following empirical relation, as illustrated in Figure 3c:  
\[\left(R/\text{S}^{*}\right)_{n}\sim n^{H} \tag{3}\]  
In log-log axes, the slope of a least-squares regression is the estimate of the Hurst exponent \(H\). The parameter \(H\) varies from 0 to 1 and has the same meaning as the \(\alpha\) parameter (Figure 3d): \(H\approx 0.5\) for the case of white noise, i.e., uncorrelated; \(H>0.5\) if the process is persistent; and for an anti-persistent case, \(H<0.5\). Moreover, it also expresses the time series' fractal dimension (\(d\)) from the relation \(d=2-H\), where 1 \(<d<2\)[44].