## 3 Data-Driven Recurrence Quantification Analysis (RQA)  
### Brief Description on Recurrence Plot (RP) and RQA  
Recurrence plot (RP) measure recurrences of a trajectory \(x_{i}\in\mathbb{R}^{m}\) in its m-dimensional phase space by the recurrence matrix \(R\), which is defined as the equation below [33]. \[R_{i,j}(\epsilon)=\Theta\big{(}\epsilon-\|x_{i}-x_{j}\|\big{)},\ i,j=1,2,\cdots,N \tag{16}\]  
where \(N\) is the number of measured points, \(\epsilon\) is a recurrence threshold distance, \(\Theta(\cdot)\) is the Heaviside function, and \(\|\cdot\|\) is a norm. Then, \(R_{i,j}(\epsilon)=1\) if the states of the system at time \(i\) and that at time \(j\) are in an \(\epsilon\)-neighborhood, and \(R_{i,j}(\epsilon)=0\) otherwise. The phase space trajectory \(x_{i}\) is reconstructed from a time series \(\{u_{i}\}_{i=1}^{N}\) by time delay embedding [33]. \[x_{i}=\Big{(}u_{i},u_{i+\tau},u_{i+2\tau},\cdots,u_{i+(m-1)\tau}\Big{)} \tag{17}\]  
where \(m\) is the embedding dimension and \(\tau\) is the time delay. The RP is obtained by plotting the recurrence matrix \(R\) in its binary entries: a black dot if \(R_{i,j}=1\) (recurrence) and a white dot if \(R_{i,j}=0\) (no recurrence). Thus, by the definition of \(R\), the RP has a black main diagonal line and is symmetric with respect to the main diagonal [33]. Since the initial purpose of RP was to visualize trajectories in the phase space, the interpretation of RP is qualitative and subjective. Therefore, it lacks quantitative standards [1]. Hence, a quantification of the obtained structures of RP is necessary for a more objective investigation of the considered system. To this end, the recurrence quantification analysis (RQA) was proposed as a quantitative description of RP by [PERSON] et al. [34], [PERSON] et al. [35], and [PERSON] et al. [36] whose diagonal and vertical lines are the base [37]. Furthermore, the RQA can provide information about different characteristics of the RP and can be used to detect transitions in dynamical systems. The RQA measures, which are adopted in this paper, are as follows:  
Recurrence rate, RR, is a measure of the density of recurrence points in the RP [37]. \[\text{RR}(\epsilon)=\frac{1}{N^{2}}\sum_{i,j=1}^{N}R_{i,j}(\epsilon) \tag{18}\]  
where \(R_{i,j}(\epsilon)\) is the recurrence matrix defined in Equation (13) and the main diagonal (\(i=j\)) is usually excluded. The RR can be used to reveal changes in dynamic systems. Determinism, DET, is the ratio of recurrence points that form diagonal structures of at least length \(l_{min}\) to all recurrence points [37]. \[\text{DET}=\frac{\sum_{l=l_{min}}^{N}IP(l)}{\sum_{l=1}^{N}IP(l)} \tag{19}\]  
Figure 3: Reconstruction error of the simulation signal. where \(P(l)\) is the histogram of diagonal lines of length \(l\) given the threshold \(\varepsilon\). Processes with uncorrelated or weakly correlated, stochastic or chaotic behavior cause none or very short diagonals, whereas deterministic processes cause longer diagonals and less single, isolated recurrence points. Therefore, the DET provides a measure for determinism or predictability of the system. Average diagonal line length, \(L\), is the average time that two segments of the trajectory are close to each other [37]. \[L=\frac{\sum_{l=l_{min}}^{N}lP(l)}{\sum_{l=l_{min}}^{N}P(l)} \tag{20}\]  
This can be interpreted as the mean prediction time. Longest diagonal line, \(L_{max}\), is related to the exponential divergence of the phase space trajectory [37]. \[L_{max}=max\Big{(}\{l_{i}\}_{i=1}^{N_{l}=l}\Big{)} \tag{21}\]  
where \(N_{l}=\sum_{l\geq l_{min}}P(l)\) is the total number of diagonal lines. The faster the trajectory segments diverge, the lower the \(L_{max}\) will be. Analogously to \(L_{max}\), the maximal length of the vertical lines, \(V_{max}\), in the RP can be regarded as [37] the following equation. \[V_{max}=max\Big{(}\{v_{l}\}_{l=1}^{N_{v}}\Big{)} \tag{22}\]  
where \(N_{v}\) is the absolute number of vertical lines.