Introduction

Recurrence Plot (RP) was initially introduced by Eckman et al. (1987) [1] as a tool for analyzing experimental time series data, especially useful for finding hidden correlations in highly complicated data and to determine the stationarity of the time series. This method allows the identification of system properties that cannot be observed by the linear and nonlinear usual approaches It is worthwhile mentioning the simplicity of the algorithms during numerical calculations too. An RP is an injective application of a single reconstructed trajectory to the boolean matrix space, each pair $y_i$, $y_j$ coming from the time series is related with a pair $(i,j)$, called recurrence points. Let us consider $N$ values of a time series given by $\{x_0, \cdots, x_{N-1}\}$, with $N$ large enough in order to evaluate the embedding dimension by using the false nearest neighbor [2] ($d \ge 2$) and the time delay ($\tau \ge 1$) by looking at the relative minimum in the mutual information [3] Following Takens' embedding theorem [4], the dynamics can be appropriately represented by the phase space trajectory reconstructed by using the time delay vectors $y_i=(x_i, x_{i+\tau}, \cdots, x_{i+(N-1) \, \tau})$. and the recurrence matrix is:

$$R_{(i,j)}=\Theta(\delta_h-\vert\vert y_i-y_j\vert\vert _{\infty}) -\Theta(\delta_l-\vert\vert y_i-y_j\vert\vert _{\infty})$$ (1)

where $\Theta$ is the Heaviside function and the matrix is symmetric. This means a RP is built by comparing all delayed vectors with each other. A dark dot is plotted, ($R_{(i,j)}=1$) with integer coordinates $(i,j)$ when $\delta_l \le \vert\vert y_i-y_j\vert\vert _{\infty} \le \delta_h$, otherwise a white dot is plotted ($R_{(i,j)}=0$). The interval $[\delta_l, \delta_h]$ it is known as threshold corridor. The choice of this interval is critical,if too large produces a saturation of the RP including irrelevant points, and if too narrow loses information. Since up to now in the literature there is no satisfactory solution, an educated guess should be appropriate. Webber and Zbilut[15] prescribe a threshold corridor corresponding to lower ten percent of the entire distance range in the corresponging unthresholded recurrence plot. In this work we use $[\sigma/10^5, \sigma/10^2]$, where $\sigma$ is the standard deviation. Webber et al. [5] in order to characterize and analyze recurrent plots introduced a set of quantifiers, which are collectively called recurrence quantified analysis (RQA). The first of this quantifiers is the % recurrence (%REC), defined as:

$$\texttt{\%REC}=100 \, \frac{N_r} {N_t}$$ (2)

where $N_t=\textrm{dim}(R)$ (every possible points) and $N_r$ is number of recursive points given by: $N_r= 2 \char93 \{ (i,j) / R_{(i,j)} > 0 \, \textrm{and} \, i<j \}$ The slope of the linear region in the S-shaped %REC vs. corridor width is the correlation dimension. The second RQA quantifier is called % determinism (%DET); and it is defined as:

$$\texttt{\%DET}=100 \, \frac{N_l}{N_r}$$ (3)

where $N_l$ is called the number of periodic dots given by: $N_l = 2 \char93 \{ (i,j) / (i,j) \in d_c(k,b), \, i<j, \forall \, c, \, k, \, b>0 \}$ and a periodic line with length $b$, origin $k$ and zone $c>0$ is defined as: $d_c(k,b) = \{ (i,i+c) / \prod_{i=k}^{k+b} R_{(i,i+c)} >0 \}$ The %DET is related with the organization of the RP. The third RQA quantifier, called entropy (S), is closely related to %DET.

$$S=-\sum^H_{b=1} P_b \log_2(P_b)$$ (4)

where $H$ is the length of the maximum periodic line, $P_n \ne 0$ is the relative frequency of the periodic lines with length $b > 0$. The label entropy is just that, a label, not to be confused with Shannon's entropy since there is not a one to one correspondence between this quantifier and the Shannon's entropy. This quantity should be labeled more properly as first rate cumulant since it is related with the relative frequency fluctuations. Moreover, for periodic orbits they are mapped onto diagonals with different lengths and uniformly distributed, giving values of $S>0$, with a maximum value. Webber assumes that $S$ is related with Shannon's entropy if and only if the system is chaotic and the embedding dimension large enough. The fourth quantifier is the longest periodic line found during the computation of %DET given by the $L_\textrm{max}$. Eckman et al. claim that line lengths on RP are directly related to inverse of the largest positive Lyapunov exponent. Short lines values are therefore indicative of chaotic or stochastic behavior.