# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 3 Applications
### Three-Point Bending Test of a Glass Fiber-Reinforced Polymeric Plate
#### 3.1.3 Discussion  
(i) Considering the DFA and Hurst parameters, there is no discernible parameter trend in Figure 7c to indicate criticality. These parameters' behavior can be clarified with the aid of Figure 8, illustrating the analysis of the exponents \(\alpha\) and \(H\) for two signal samples, for instants 0.24 and 0.82. There are two discernible regions for each dataset, and the blue lines show their corresponding linear fits. However, this effect is masked if a single fit is calculated for the whole dataset (red line). This is a crucial difference because the two regions indicate different behaviors concerning time correlation: short-term correlations are persistent, whereas long-term ones are anti-persistent. Similar effects can be found in other works, such as [PERSON] et al., 2003 [54], [PERSON] et al., 2014 [66], and [PERSON] et al., 2004 [67]. To further explore this behavior, \(\alpha\) was re-evaluated for short-term windows (\(n<300\)) and long-term ones (\(n\geq 300\)), respectively designated \(\alpha_{short}\) and \(\alpha_{high}\). The corresponding results are depicted in Figure 9, where a 100-point moving mean was added to (dotted line) improve tendency visualization in each region, in the same figure the load evolution (\(\rm F/F_{max}\)) during the simulation in black line is presented. Figure 8: Details of the Hurst and DFA exponents for (**a**) signal at normalized time 0.24; (**b**) signal at normalized time 0.82. The correlation force for the short-term windows tends to be persistent throughout the entire test, increasing significantly between the normalized instants 0.78 and 0.88, which correspond to intense AE activity as the structure approaches collapse. During the same period, long-term correlation forces are nearly nil. This opposite convergence of \(\alpha_{short}\) and \(\alpha_{high}\) may indicate spasmodic behavior. A similar effect is also observed between instants 0 and 0.24: initially, both \(\alpha_{short}\) and \(\alpha_{high}\) have an approximated average of 0.7 until normalized instant 0.18. Then, \(\alpha_{high}\) falls within the noncorrelated range (<0.5) that precedes an avalanche (see Figure 7a) with large-amplitude signals. (ii) Regarding the natural time parameters that are presented in Figure 7d, it is necessary to check whether the identified critical points represent \"true coincidence\", i.e., not only are all the criteria satisfied, but they are also invariant to the selected threshold. Applying the same methodology that is described in the previous section, two new threshold values are applied to separate hits from the overall AE signals: those with amplitudes that are higher than (a) 1 mV and (b) 2 mV. In the first case, the reconstructed time series comprises of 2832 events, whereas the second one has 1164. The corresponding results in natural time are given in Figure 10. Compared with Figure 7d, two critical points are confirmed for all the thresholds: the ones approximately at normalized times 0.27 and 0.59. As for the third possibly critical point that is identified in Figure 7c, it is not an actual critical point since it does not appear for the other two thresholds. As mentioned earlier, the first critical state (0.27) is related to a dense region of AE signals between 0.42 to 0.52 of the normalized time, see Figure 7a,b, where the stiffness of the structure is affected as shown by the instability that is present in the load curve (see Figure 7a). The second point (0.59) indicates the critical stage preceding structure failure, which occurs about 30 s later. Additionally, \(\kappa_{1}\) tends to stabilize at nearly 0.07 for normalized time 0.6, regardless of the selected three thresholds value for hit identification. That means that the system has started to self-organize, an obvious prerequisite for failure [68]. Figure 9: DFA considering short-term (\(\alpha_{short}\)) and long-term (\(\alpha_{high}\)) windows for the GFRP plate. The load and AE signals out of scale.