Recurrence Plots and Cross Recurrence Plots

Variations of Recurrence Plots

In the original definition of the RPs, the neighbourhood is a ball (i.e. \(L_2\)-norm is used) and its radius is chosen in such a way that it contains a fixed amount of states \(\vec{x}_j\) (Eckmann et al., 1987). With such a neighbourhood, the radius \(\varepsilon_i\) changes for each \(\vec{x}_i\) (\(i=1,\ldots,N\)) and \(R_{i,j} \not= R_{j,i}\) because the neighbourhood of \(\vec{x}_i\) does not have to be the same as that of \(\vec{x}_j\). This property leads to an asymmetric RP, but all columns of the RP have the same recurrence density. We denote this neighbourhood as fixed amount of nearest neighbours (FAN).

However, the most commonly used neighbourhood is that with a fixed radius \(\varepsilon_i=\varepsilon,\ \forall i\). For RPs this neighbourhood was firstly used by Zbilut et al. (1991). A fixed radius means that \(R_{i,j} =R_{j,i}\) resulting in a symmetric RP. The type of neighbourhood that should be used depends on the application. Especially in applications of the cross recurrence plots, the neighbourhood with a FAN will play an important role.

In the literature further variations of the recurrence plots have been proposed:

Variations of recurrence plots
Examples of various defined RPs for a section of the \(x\)-component of the Lorenz system (sampling time \(\Delta t = 0.03\)): (A) RP computed by using the \(L_{\infty}\)-norm, (B) RP computed by using the \(L_1\)-norm, (C) RP computed by using the \(L_2\)-norm, (D) RP computed by using a fixed amount of nearest neighbours (FAN), (E) RP computed by using a threshold corridor \([\varepsilon_\text{in},\varepsilon_\text{out}]\), (F) perpendicular RP (\(L_2\)-norm), (G) distance plot (unthresholded RP, \(L_2\)-norm), (H) order patterns RP (\(m = 3, \tau_{1, 2, 3} = 9\)) and (I) RP, where the \(y\)-axis represents the relative time distances to the next recurrence points but not their absolute time (“close returns plot”, \(L_2\)-norm). Except for (F), (G) and (H), the parameter \(\varepsilon\) is chosen in such a way that the recurrence point density (RR) is approximately the same. The embedding parameters (\(m=5\) and \(\tau=5\)) correspond to an appropriate time delay embedding. Please click in the figure to find out the differences in detail!

The selection of a specific variant from this variety of RPs depends on the problem and on the kind of data. Perpendicular RPs are highly recommended for the quantification analysis based on diagonal structures, whereas corridor thresholded RPs are not suitable for this task. Windowed RPs are appropriate for the visualization of the long range behaviour of rather long data sets. If the recurrence behaviour for the states \(\vec{x}_i\) within a predefined section \(\{\vec{x}_{i+i_0},\ldots,\vec{x}_{i+i_0+k}\}\) of the phase space trajectory is of special interest, an RP with a horizontal LOI will be practical.

It should be emphasized again that the recurrence of states is an important feature. Beside the recurrence plots, there are some other methods that use recurrences. For example, the recurrence in the phase space is used for the recurrence time statistics (Kac, 1947; Gao, 1999; Balakrishnan et al., 2000), first return map (Lathrop and Kostelich, 1989), space time separation plot (Provenzale et al., 1992) or as a measure for nonstationarity (Kennel, 1997; Rieke et al., 2002; closely related to the recurrence time statistics).

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