## Appendix C Recurrence Quantification Analysis  
RQA is a statistical, graphical, and analytical tool [33; 34] that has been successfully used in several fields, such as finance [35], physiology [36; 37; 38; 39; 40], earthquakes, and geoscience [41; 42]. It is useful for analysing nonlinear dynamic systems. It has been demonstrated that RQA is a reliable tool for analysing the stochastic and chaotic dynamics of physiological signals [43; 44]. The steps for the RQA analysis are explained below:  
* Phase Space Reconstruction: the time series data is often reconstructed into a phase space to reveal the underlying dynamics. This is done using delay embedding techniques. Let's supposed to have a time series \(x(t)\), where \(t=1,2,\ldots,N\). The phase space can be reconstructed by creating a set of vectors \(Y(i)\), where each vector is given by: \[Y(i)=[x(i),x(i+\tau),x(i+2\tau),\ldots,x(i+(m-1)\tau)]\] (1) Here, m is the embedding dimension, and \(\tau\) is the time delay, both of which need to be appropriately chosen. * Constructing the Recurrence Matrix: A matrix R is then constructed by comparing each point in the phase space with every other point. The elements of the matrix R are defined by: \[Ri,j=\Theta(\epsilon-||\ Y(i)-Y(j)\ ||)\] (2) where \(\Theta\) is the Heaviside step function, \(\epsilon\) is a predefined threshold distance (or radius), and \(||\cdot||\) denotes a norm (often the Euclidean norm). \(Rij\) has value 1 if the distance between \(Y(i)\) and \(Y(j)\) is less than \(\epsilon\); otherwise, its value is 0. The recurrence matrix is then obtained: \[ \begin{bmatrix}RR_{1}&RR_{3}&\dots\\ RR_{2}&RR_{4}&\dots\\ \dots&\dots&\dots\\ RR_{N}&\dots&\dots\end{bmatrix} = \begin{bmatrix}e_{1,1}&e_{1,2}&\dots&e_{1,m}\\ e_{2,1}&e_{2,2}&\dots&e_{2,m}\\ \dots&\dots&\dots&\dots\\ e_{N-m+1,1}&e_{N-m+1,2}&\dots&e_{N-m+1,m}\end{bmatrix} \] (11)
* Visualizing the Recurrence Plot: The recurrence plot is a two-dimensional representation of the matrix \(R\). Each point (\(i\),\(j\)) is plotted as a small dot if \(Ri\),\(j\) = 1. The result is a plot that reveals patterns of recurrence within the dynamical system. Diagonal lines in the plot, for example, indicate a repeating pattern or periodic behaviour (Figure 10). The choice of parameters like \(\epsilon\), \(\tau\), and m is crucial, as it can significantly affect the structure and interpretation of the RP. Typically, these parameters are chosen based on the specific characteristics of the time series being analyzed and the objectives of the study. For our study, the following RQA parameters were used and held constant: line = 4, embedding dimension = 7, radius or cut-off = 5. For further details about parameters optimization procedure, see [6]. RQA involves quantifying the patterns observed in the recurrence plot. Several measures are used to describe the patterns, including:  
* Recurrence Rate (REC): This is the simplest RQA measure and is defined as the fraction of points in the recurrence plot that are recurrences. * Determinism (DET): This measure reflects the predictability of the system. It is the fraction of recurrence points that form diagonal lines of at least a minimum length \(\mathrm{l}_{\mathrm{min}}\) in the RP. * Laminarity (LAM): This measure reflects the proportion of recurrence points that form vertical (or horizontal, depending on the orientation) lines of at least a minimum length \(\mathrm{v}_{\mathrm{min}}\) in the RP. The choice of the minimum line length is crucial and depends on the specific application or the characteristics of the data being analyzed. These measures provide a quantitative way to describe the patterns observed in a recurrence plot, offering insights into the dynamics of the system under study. They are particularly useful in identifying and characterizing complex, nonlinear behaviours in time series data. RQA is used in a wide range of fields including physics, biology, finance, and environmental sciences. It is particularly useful in analysing complex, nonlinear, or non-stationary time series data.