## 2 Materials and Methods
### Rqa
#### 2.4.1 Phase Space Reconstruction  
The trajectory in phase space represents the time development and system dynamics. With defined variables, every system is possible to describe. These variables form the trajectory in time and n-dimensional space, or precisely in the phase space [34]. In many cases, it is not possible to record or measure all the variables of the system's state. The phase space, however, based only on one state variable, might be constructed. A widely and most frequently used method is an embedding dimension and time delay method by Dutch mathematician [PERSON], see equation below [34]. \[x_{i}=(u_{i},u_{i+\tau},...,u_{i+(m-1)\tau})^{T} \tag{1}\]  
where \(u\) is observed state variable, \(\tau\) is time delay and \(m\) is inserted dimension. The reconstructed phase space does not entirely match the original phase space, but in case that the inserted dimension is sufficiently large, its topological properties remain the same [34]. The inserted dimension should be at least twice the size of the attractor's dimension, precisely \(m>2d+1\)[34; 35]. There are several opinions on the input parameters (dimension, time delay) setup. The optimal setup of these parameters is important when reconstructing the phase space, which will fully depict system dynamics. Time delay sets the distance between neighboring elements. If the value of time delay was too big, the states of reconstructed phase space might appear independent, and thus the reconstructed trajectory will seem to be a random process. In contrast, if the value of time delay was small, the difference between individual states of the reconstructed phase space will be negligible. In the past, to choose time delay, an autocorrelation function was used. Unfortunately, this method did not take into account the possibility of nonlinear processes [35]. Recent studies indicate that the method convenient for choosing the specific time delay is the mutual information [35; 36]. The mutual information represents the information of mutual dependency of two dependent parameters. Higher mutual information means that the variables are more dependent on each other. The most suitable length of time delay is the first minimum value of mutual information. During the monitoring \(x(t_{i}+\tau)\) in time \(t_{i}+\tau\), on average the first minimum value presents the highest information contribution with regard to the information from the monitoring \(x(t_{i})\) in time \(t_{i}\). The mutual information of two variables \(A\) and \(B\) might be defined through the entropy as:  
\[I(A,B)=H(A)+H(B)-H(A,B) \tag{2}\]  
where \(H(A)\) and \(H(B)\) are entropy and \(H(A,B)\) is joint entropy of \(A\) and \(B\). After the selection of the optimal time delay, it is necessary to choose the optimal dimension of an embedment. A frequently used method for selecting the optimal embedded dimension is a method of false nearest neighbors, and that is because of the fact that after projecting a system trajectory of the original phase space into space with low-dimension, there is a self-crossing of the trajectory. Because of the self-crossing, the false nearest neighbors, the number of which diminishes as the dimension increases, are formed. A disadvantage of using this method is that a threshold must be set to consider neighbors as false. An interesting modification of false nearest neighbors method that eliminates the mentioned disadvantage was described by [PERSON] [37], who used the Euclidean quotient distances of two closest neighboring states in the dimension \(m\) and dimension \(m+1\), see Formula (3). \[a(i,m)=\frac{||y_{i}(m+1)-y_{n(i,m)}(m+1)||}{||y_{i}(m)-y_{n(i,m)}(m)||} \tag{3}\]  
where \(||\) is Euclidean distance, \(y_{i}(m)\) is the i-reconstructed vector of \(m\) and \(y_{n(i,m)}\) is the closest neighbor \(y_{i}(m)\). Furthermore, a variable called the average of all value \(E(m)\) is introduced [37]:  
\[E(m)=\frac{1}{N-m\tau}\sum_{i=1}^{N-m\tau}a(i,m) \tag{4}\]The average \(E(m)\) is dependent on dimension \(m\) and time delay \(\tau\).