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:

• Iwanski and Bradley (1998) have defined a variation of an RP with a corridor threshold $$[\varepsilon_{\text{in}}, \varepsilon_{\text{out}}]$$, $$R_{i,j} = \Theta (\| \vec{x}_i - \vec{x}_j\| - \varepsilon_{\text{in}}) \cdot \Theta (\varepsilon_{\text{out}} - \| \vec{x}_i - \vec{x}_j\|).$$ Those points $$\vec{x}_j$$ are considered to be recurrent that fall into the shell with the inner radius $$\varepsilon_\text{in}$$ and the outer radius $$\varepsilon_\text{out}$$. The authors have suggested to use this kind of RPs in order to study “interesting structures” in the RP. An advantage of such a corridor thresholded recurrence plot is its increased robustness against recurrence points coming from the tangential motion. However, the threshold corridor removes the inner points in broad diagonal lines, which results in two lines instead of one. These RPs are, therefore, not suitable for a quantification analysis. The usage of a shell as a neighbourhood can be found in an algorithm for computing Lyapunov exponents from experimental time series (Eckmann et al, 1986).

• Choi et al. (1999) have introduced the perpendicular recurrence plot $$R_{i,j} = \Theta \left( \varepsilon - \| \vec{x}_i - \vec{x}_j \|\right) \cdot \delta \left(\dot{\vec{x}}_i ( \vec{x}_i -\vec{x}_j)\right).$$

Here, $$\delta$$ is the Delta function. This recurrence plot contains only those points $$\vec{x}_j$$ that fall into the neigbourhood of $$\vec{x}_i$$ and lie in the $$(m-1)$$-dimensional subspace that is perpendicular to the phasespace trajectory at $$\vec{x}_i$$. These points correspond locally to those lying on a Poincaré section. This criterion cleans up the RP more from recurrence points based on the tangential motion than the previous corridor thresholded RPs. The authors have shown the increased efficiency of the perpendicular RPs for their application on estimation of the largest Lyapunov exponent. Using this kind of an RP, the finding of unstable periodic orbits (if they exist) is more robust.

• In a similar direction goes the iso-directional recurrence plot, introduced by Horai et al. (2002), $$R_{i,j} = \Theta \left( \varepsilon - \| (\vec{x}_{i+T} - \vec{x}_i) - (\vec{x}_{j+T} - \vec{x}_j) \|\right).$$ Such recurrence points are related with neighboured trajectories which run parallel and in the same direction. Horai introduced an additional iso-directional neighbours plot, which is simply the product between the common recurrence plot and the iso-directional recurrence plot $$R_{i,j} = \Theta \left( \varepsilon - \| \vec{x}_i - \vec{x}_j \|\right) \cdot \Theta \left( \varepsilon - \| (\vec{x}_{i+T} - \vec{x}_i) - (\vec{x}_{j+T} - \vec{x}_j) \|\right).$$ The computation of this special recurrence plot is simpler than that of a perpendicular recurrence plot. Although the cleaning the RP from false recurrences is better than in a common recurrence plot, it does not reach the quality of a perpendicular recurrence plot. A disadvantage is the additional parameter $$T$$ which has to be thoughtfully determined before (however, it seems that this parameter has to be related with the embedding delay $$\tau$$).

• The RP contains, finally, tests of all states with each other, which results in $$N^2$$ tests for $$N$$ considered states. Still, it is also possible to test each state with a predefined amount $$k$$ of subsequent states (Zbilut et al., 1991; Koebbe and Mayer-Kress, 1992; Atay and Altintas, 1999) $$R_{i,j} = \Theta ( \varepsilon - \| \vec{x}_i - \vec{x}_{i-i_0+j-1} \|), \qquad i=1,\ldots, N-k, \quad j=1,\ldots,k.$$ This reveals an $$(N-k) \times k$$-matrix which does not have to be square. The $$y$$-axis represents the time distances to the following recurrence points but not their absolute time. All diagonal oriented structures in the common RP are now projected to the horizontal orientation. For $$i_0=0$$, the LOI, which was the diagonal line in the common RP, is now the horizontal line on the $$x$$-axis. With non-zero $$i_0$$ the RP contains recurrences of a certain state only in the predefined time interval after time $$i_0$$ (Koebbe and Mayer-Kress, 1992).

This representation of recurrences may be more intuitive than the RPs usually are because the consecutive states are not oriented diagonally. However, such an RP represents only the first $$(N-k)$$ states. Mindlin and Gilmore (1992) have proposed the close returns plot which is, in fact, such an RP exactly for one dimension. Using this kind of RP, a first quantification approach of RPs (or “close returns plots”) can be found (“close returns histogram”, recurrence times). It has been used for the investigation of periodic orbits and topological properties of strange attractors (Lathrop and Kostelich, 1989;Tufillaro et al. 1990; Mindlin and Gilmore, 1992).

• Instead plotting the recurrences with black points, the distances $$D_{i,j} = \| \vec{x}_i - x_j\|$$ between the states $$\vec{x}_i$$ and $$\vec{x}_j$$ can be plotted. Although this is not a real recurrence plot, it is sometimes called global recurrence plot (Webber Jr., 2003) or unthresholded recurrence plot (Iwanski and Bradley, 1998). However, it should be termed distance plot. This representation can also help in studying phase space trajectory. Moreover, it may help to find an appropriate threshold value $$\varepsilon$$.

• The windowed and meta recurrence plots have been suggested as means of investigating an external force or the nonstationarity in a system (Manuca and Savit, 1996; Casdagli, 1997). The first ones are obtained by covering an RP with $$w \times w$$-sized squares (windows) and by averaging the recurrence points that are contained in these windows (Casdagli, 1997). Consequently, a windowed recurrence plot is an $$N_w \times N_w$$-matrix, where $$N_w$$ is the floor-rounded $$N/w$$, and consists of values which are not limited to zero and one (this suggests a colour-encoded representation). These values correspond with the cross correlation sum $$C_{I,J} = \frac{1}{w^2} \sum_{i,j=1}^w R_{i+(I-1)w,j+(J-1)w} \qquad I,J = 1,\ldots,N/w$$ between sections in $$\vec{x}$$ with length $$w$$ and starting at $$(I-1)w+1$$ and $$(J-1)w+1$$ (for cross-correlation integral cf. Kantz and Schreiber 1994). The meta recurrence plot as it has been defined by Casdagli (1997) is a distance matrix derived from the cross correlation sum, $$D_{I,J} = \frac{1}{\varepsilon^m} \left(C_{I,I} + C_{J,J} - 2 C_{I,J}\right).$$ By applying a further threshold value to $$D_{I,J}$$ (analogous to recurrence plots), a black-white dotted representation is also possible.

Manuca and Savit (1996) have gone one step further. They have used quotients from the cross correlation sum to form a meta phase space. From this meta phase space a recurrence or non-recurrence plot is created, which can be used to characterize the nonstationarity in time series. For a sufficient explanation the work of Manuca and Savit (1996) is recommended.

• Instead of using the spatial closeness between phase space trajectories, order patterns recurrence plots use order patterns $$\pi$$ for the definition of a recurrence. An order pattern $$\pi$$ of dimension $$m$$ is defined by the discrete order sequence of the data series $$x_i$$ and has length $$m$$. For $$m = 3$$ we have, e. g., six order patterns: Using order patterns, the data series $$x_i$$ will be symbolized by order pattern:

$$x_i, x_{i-\tau_1},\ldots,x_{i-\tau_{m-1}} \rightarrow \pi_i$$ The order patterns recurrence plot is then defined by the pairwise test of order patterns (Groth, 2005): $$R_{i,j} = \delta ( \pi_i, \pi_j).$$ Such a recurrence plot represent those times, when specific rank order sequences in the system recur. Its main advantage is its much better robustness against non-stationary data.

• Instead of using the spatial closeness between phase space trajectories, isometric recurrence plots (Sabelli, 2001) test only for similar lengths of phase space vectors $$R_{i,j} = \Theta \left( \varepsilon - \left| \|\vec{x}_i\| - \|\vec{x}_j\| \right|\right).$$ This kind of recurrence was proposed for the study of "bios". However, it is very sensitive to changes in the absolute amplitude (e.g., a normalisation to mean zero would change such recurrence). Therefore, it is also very sensitive to noise. Such an RP contains much more recurrence points than the other variants. Moreover, many diagonal lines with slopes –45° will appear, indicating non-causal recurrences. Therefore, this kind of RP is not suggested. 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). © 2000-2019 SOME RIGHTS RESERVED 