# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 2 Theoretical Background
### Long-Range Correlations Measures (DFA and Hurst)  
Complex systems appear in many sciences, in topics as varied as heart rate dynamics, DNA, neuron spiking, meteorology, human motor activity, seismic signals, and economic time series [31; 32]. These systems are characterized by distinct effects such as nonlinearity, spontaneous order, and adaptation, among others, and can be studied through fractal properties [33; 34]. In AE signals such as those in Figure 2, interpreting events as irregularities in a time series, their occurrence can be identified through the series properties, such as self-similarity and scale-invariance. For instance, long-range power-law correlations are present if a time series exhibits scale-invariant features [35; 36]. The strength or persistence of such long-range correlations helps to unravel or explain data behavior [35]. There are several methods to assess and quantify the strength of these correlations. This work uses two: detrended fluctuation analysis (DFA) and rescaled range analysis (\(R\)/S*). Figure 1: The \(b\)-value evolution and its physical meaning with respect to an acoustic emission signal. The DFA is a method that was initially proposed by [PERSON] et al., 1994 [PERSON] et al. (1994) to detect long-range correlations in time series with non-stationaries [34; 36; 38; 39; 40]. Briefly, DFA fits a power law for estimating \(\alpha\), which is a measure of long-range correlations. It is calculated by plotting in log-log axes the average fluctuations in the series, named \(F_{\text{DFA}}\), over different sampling scales (window sizes): \(\alpha\) corresponds to the angular coefficient in the linear fit of \(F_{\text{DFA}}\) versus windows size. Figure 2 exemplifies the application of the DFA method for a time series in the form \(\{x(i)\}_{i=1}^{N_{\text{max}}}\), where \(N_{\text{max}}\) is the total number of points in the series, in the case of a signal series \(x(i)\) with \(N_{\text{max}}\) = 1000, Figure 2a. The general steps for applying the DFA are as follows [23; 34; 41]:  
1. Integrate the series \(x(i)\), obtaining \(y(k)=\sum_{i=1}^{k}x(i)-\langle x\rangle\), where \(k=1,\ldots,N_{\text{max}}\) and \(\langle x\rangle\) stands for the average of \(x\), see Figure 2b. 2. Divide \(\{y(k)\}_{i=1}^{N_{\text{max}}}\) into non-overlapping windows of equal length \(n\). 3. For each \(n\)-size window, fit a least-squares line to the integrated signal \(y(k)\) (red line in Figure 2b). The sequence of fitted lines constitutes the trend series \(y_{n}(k)\). Figure 2b shows the integrated series with three different window sizes, \(n\) = 250, \(n\) = 100, \(n\) = 50. 4. Calculate the average fluctuation \(F_{\text{DFA}}\) of the integrated series \(y(k)\) around the trend series \(y_{n}(k)\). Explicitly, \(F_{\text{DFA}}(n)=\sqrt{(1/N_{\text{max}})\sum_{k=1}^{N_{\text{max}}}[y(k)-y_{ n}(k)]^{2}}\). 5. Repeat step 4 for a broad range of scales (sizes of \(n\)) to provide a relationship between \(F_{\text{DFA}}\) and \(n\), i.e., \(F_{\text{DFA}}(n)\). Figure 2: Graphical illustration of the DFA steps. (**a**) Example of time series; (**b**) integrated series divided into different size windows with their corresponding linear-fit trends; (**c**) log-log plot of average fluctuations \(F_{\text{DFA}}(n)\) versus window sizes \(n\). The estimated scaling exponent \(\alpha\) is the slope of the least-squares fitted line; (**d**) representative examples of white (random), pink (fractal, long-range correlated, slightly persistent), and blue (anti-persistent) time and power-spectrum series. (Adapted from ref. [37]). Plot \(\log\)\(\text{F}_{\text{DFA}}(n)\) versus \(\log(n)\) (see Figure 2c). If there is an obvious linear relationship between them, the slope of its least-squares fit estimates the scaling exponent \(\alpha\). As explained in detail in [34; 37], applying the DFA methodology yields both a measure of persistence and an index of self-similarity in the time series.