# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 3 Applications
### \"Spaghetti\" Bridge Model
#### 3.2.2 Results  
The \"Spaghetti\" bridge model results throughout the loading test are shown in Figure 14. For a threshold level of 0.7 mV, the AE sensors detected 230 signals, Figure 14a. These results appear in Figure 14b as the numbers of instantaneous and accumulated signals, the loading evolution (\(\mathrm{F}/\mathrm{F}_{\mathrm{max}}\)), and the acoustic emission energy (\(\mathrm{E}_{\mathrm{AE}}/\mathrm{E}_{\mathrm{AEmax}}\)) against the normalized time. For this test, signals occur nearly when the load increases, indicating that their distribution depends explicitly on the loading pattern. Additionally, up to \(t\approx 0.6\), signal counts grow at an approximately constant rate. Then, there is a sharp increase in the number of signals for a brief period. After that, the signal counts grow once again at a nearly constant rate but a faster pace. That agrees with the expected general behavior for AE-based analysis methods, and this work seeks to confirm such a conclusion through the methods that are presented in Section 2, as summarized in Figure 14.  
The \(b\)-value parameter (Figure 14a) was calculated in [27] by separating the dataset into packs of 25 signals, with five-signal overlaps between the successive packs. The sudden coefficient changes at \(t\approx 0.6\) coincide with the avalanche that is observed in the signals count, confirming this point as critical for the test in question.  
The long-range correlation results for the spaghetti bridge appear in Figure 14c. The calculation methodology is the same one that is used with the GFRP plate, but since this test has yielded fewer signal samples, a 6-signals moving window was used to compute \(\alpha\) and \(H\). Both parameters behave similarly until \(t\approx 0.55\), with values of about 0.9 indicating persistent correlations. Then, they present significant local changes, most notably at normalized instants 0.6, 0.65, and 0.79, with the Hurst exponent appearing as more sensitive to the observed signal changes. Contrary to the GFRP plate case, no anti-persistent correlations are observed, just strong long-range correlations with local instabilities for the points of more significant AE activity.  
Finally, the results for natural time analysis are presented in Figure 14d, considering \(Q_{k}=A_{\mathrm{max},k}\)[15] and using the same calculation method that was described for the GFRP plate. A single coincidence point is identified at a normalized time 0.1206. If only the order parameters for criticality (\(\kappa_{1}\) and \(\langle D\rangle\)) are considered, new convergence points are observed at the normalized instant 0.54, which is close to the already discussed sudden rise in AE energy at \(t\approx 0.6\) (Figure 14b), which was also identified through the \(b\)-value (Figure 14a). The last convergence point occurs at \(t=0.62\), where the two exponents change behavior  
Figure 13: “Spaghetti” Bridge Model: (**a**) Geometric restrictions for the contest, (**b**) The initial and final configurations of the bridge during the test (Adapted from ref. [27]).  
again until the structure collapses. Subsequent analysis can show that these results do not change with respect to the amplitude threshold for identifying hits. Contrary to the GFRP plate, however, there is no discernible coincidence point in the vicinity of the structure's failure.  
Figure 14: **(a)** The AE signals and corresponding \(b\)-values for the spaghetti bridge loading test, as described in [27]); **(b)** The numbers of instantaneous signals, the number of accumulated signals, the applied load, and the accumulated energy versus normalized time. The first was multiplied by 7 to maintain the same vertical scale for instantaneous and accumulated signals; **(c)** The application of Hurst and DFA methods to the AE data; **(d)** NT analysis of the AE time series (this figure’s format follows that of Figure 7d).