# Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals
## 2 Theoretical Background
### Natural-Time Analysis  
The analysis in the natural time domain allows identifying when a complex system enters a critical state [47]. For a time series comprising \(N\) events, the natural time \(\chi_{k}\) of the \(k\)-th event is defined as this event normalized, divided by the total number of the considered events, \(\chi_{k}=k/N\)[2], see Figure 4. On the other hand, natural time \(\chi_{k}\) serves as an index for the occurrence of the \(k\)-th event, Figure 4. It is, therefore, smaller than, or equal to, unity (0,1) [44,48]. We denote by \(Q_{k}\) a quantity proportional to the energy of the individual \(k\)-th event, so its normalized energy is given by \(p_{k}=Q_{k}/\sum_{n=1}^{N}Q_{n}\). Inthe specific case of AE, \(Q_{k}\) could be the signal energy measured in defined form or also the signal magnitude that will relate with the signal energy. In natural time analysis, the evolution of the pair \((\chi_{k},p_{k})\) is considered [44], see Figure 4.  
The natural time \(\chi_{k}\)'s variance \(\kappa_{1}\) is given by:  
\[\kappa_{1}=\left\langle\chi^{2}\right\rangle-\left\langle\chi\right\rangle^{2 }\equiv\sum_{k=1}^{N}p_{k}\left(\frac{k}{N}\right)^{2}-\left(\sum_{k=1}^{N} \frac{k}{N}p_{k}\right)^{2} \tag{4}\]  
\(\kappa_{1}\) varies when a new \(k\)-th event occurs, once natural time \(\chi_{k}\) changes from \(k/N\) to \(k/(N+1)\) and \(p_{k}\) changes to \(Q_{k}/\sum_{i=1}^{N}Q_{i}\). The called order parameter \(\kappa_{1}\) may identify the approach of the dynamical system to a critical point [48; 49]. Moreover, the entropy (\(S\)) in natural-time domain is defined as,  
\[S=\left\langle\chi\ln\chi\right\rangle-\left\langle\chi\right\rangle\ln\left \langle\chi\right\rangle\equiv\sum_{k=1}^{N}p_{k}\chi_{k}\ln\chi_{k}-\left( \sum_{k=1}^{N}p_{k}\chi_{k}\right)\ln\left(\sum_{k=1}^{N}p_{k}\chi_{k}\right) \tag{5}\]  
Notice that the entropy in natural time refers to numerical signal data and does not directly relate to the property from traditional thermodynamics [50], being related to the information theory instead [51]. Similar to the variance \(\kappa_{1}\), the entropy \(S\) is also a dynamic parameter depending on the sequential order of events [52]. The third variance term that is considered in this method is the time-reversal entropy \(S_{rev}\). It is obtained by reversing the order of the energy data (\(Q_{k}\)), calculating \(S\) with this new series, and then subtracting the entropy from the original series from the result. According to [PERSON] et al., 2020 [53], this entropy parameter is the most sensitive one to the ordering of the signal data. It is possible to evaluate another order parameter, which describes the \"average\" distance \(\left\langle D\right\rangle\) between the curves of the normalized power spectrum of the events [48],  
\[\Pi(\omega)=\left|\sum_{k=1}^{N}p_{k}e^{i\omega\frac{k}{N}}\right|^{2} \tag{6}\]  
where \(\omega=2\pi\phi\) (\(\phi\in(0,0.5)\) standing for the frequency in natural time) and the ideal normalized power spectrum, that is the normalized power spectrum when the sequence is in critical regime,  
\[\Pi(\omega)_{ideal}=\frac{18}{5\omega^{2}}-\frac{6\cos\omega}{5\omega^{2}}- \frac{12\sin\omega}{5\omega^{3}} \tag{7}\]  
Figure 4: Natural time in the context of acoustic emission time series (adapted from ref. [48]). if \(\omega\to 0\), Equation (7) simplifies to \(\Pi(\omega)_{ideal}\approx 1-0.07\omega^{2}\). When \(\Pi(\omega)\) approaches \(\Pi(\omega)_{ideal}\) from below, the critical state is indicated. The application of the natural time method to analyze AE signals can be described with aid from Figure 5. First, the events/hits are separated from the overall signal, Figure 5a. Then, they are ordered in the natural time domain, that is, in terms of \(\chi_{k}\) and \(Q_{k}\) (Figure 5b). Next, of the order parameters, \(\kappa_{1}\) and \(\langle D\rangle\), and the entropy measurements \(S\) and \(S_{rev}\) are calculated with Equations (4)-(7).