Recurrence Plots and Cross Recurrence Plots

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» Wolfram Demos: (1) Frequency Distribution of the Logistic Map, (2) Recurrence-Based Representations of the Logistic Map, (3) Recurrence Network Measures for the Logistic Map

Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles), but also irregular cyclicities (as El Niño Southern Oscillation). Moreover, the recurrence of states, in the meaning that states are arbitrary close after some time, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems. The recurrence of states in nature has been known for a long time and has also been discussed in early publications (e.g. recurrence phenomena in cosmic-ray intensity, Monk, 1939).

Eckmann et al. (1987) have introduced a tool which can visualize the recurrence of states \(\vec{x}_i\) in a phase space. Usually, a phase space does not have a dimension (two or three) which allows it to be pictured. Higher dimensional phase spaces can only be visualized by projection into the two or three dimensional sub-spaces. However, Eckmann's tool enables us to investigate the \(m\)-dimensional phase space trajectory through a two-dimensional representation of its recurrences. Such recurrence of a state at time \(i\) at a different time \(j\) is marked within a two-dimensional squared matrix with ones and zeros dots (black and white dots in the plot), where both axes are time axes. This representation is called recurrence plot (RP). Such an RP can be mathematically expressed as $$ R_{i,j}=\Theta(\varepsilon_i - \|\vec{x}_i - \vec{x}_j\|), \qquad \vec{x}_i \in \mathbb{R}^m, \quad i,j = 1,\ldots, N, $$ where \(N\) is the number of considered states \(x_i\), \(\varepsilon_i\) is a threshold distance, \(\| \cdot \|\) a norm and \(\Theta( \cdot )\) the Heaviside function.

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Structures in Recurrence Plots

The initial purpose of RPs is the visual inspection of higher dimensional phase space trajectories. The view on RPs gives hints about the time evolution of these trajectories. The advantage of RPs is that they can also be applied to rather short and even nonstationary data.

The RPs exhibit characteristic large scale and small scale patterns. The first patterns were denoted by Eckmann et al. (1987) as typology and the latter as texture. The typology offers a global impression which can be characterized as homogeneous, periodic, drift and disrupted.

• Homogeneous RPs are typical of stationary and autonomous systems in which relaxation times are short in comparison with the time spanned by the RP. An example of such an RP is that of a random time series.
• Oscillating systems have RPs with diagonal oriented, periodic recurrent structures (diagonal lines, checkerboard structures). For quasi-periodic systems, the distances between the diagonal lines are different. However, even for those oscillating systems whose oscillations are not easily recognizable, the RPs can be used in order to find their oscillations.
• The drift is caused by systems with slowly varying parameters. Such slow (adiabatic) change brightens the RP's upper-left and lower-right corners.
• Abrupt changes in the dynamics as well as extreme events cause white areas or bands in the RP. RPs offer an easy possibility to find and to assess extreme and rare events by using the frequency of their recurrences.

The closer inspection of the RPs reveals small scale structures (the texture) which are single dots, diagonal lines as well as vertical and horizontal lines (the combination of vertical and horizontal lines obviously forms rectangular clusters of recurrence points).

• Single, isolated recurrence points can occur if states are rare, if they do not persist for any time or if they fluctuate heavily. However, they are not a unique sign of chance or noise (for example in maps).
• A diagonal line \(R_{i+k, j+k} = 1\) (for \(k=1,\ldots,l\), where \(l\) is the length of the diagonal line) occurs when a segment of the trajectory runs parallel to another segment, i.e. the trajectory visits the same region of the phase space at different times. The length of this diagonal line is determined by the duration of such similar local evolution of the trajectory segments. The direction of these diagonal structures can differ. Diagonal lines parallel to the LOI (angle \(\pi\)) represent the parallel running of trajectories for the same time evolution. The diagonal structures perpendicular to the LOI represent the parallel running with contrary times (mirrored segments; this is often a hint for an inappropriate embedding). Since the definition of the Lyapunov exponent uses the time of the parallel running of trajectories, the relationship between the diagonal lines and the Lyapunov exponent is obvious.
• A vertical (horizontal) line \(R_{i,j+k} = 1\) (for \(k=1,\ldots,v\), where \(v\) is the length of the vertical line) marks a time length in which a state does not change or changes very slowly. It seems, that the state is trapped for some time. This is a typical behaviour of laminar states (intermittency).

These small scale structures are the base of a quantitative analysis of the RPs.

Summarizing the last mentioned points we can establish the following list of observations and give the corresponding qualitative interpretation:

The visual interpretation of RPs requires some experience. The study of RPs from paradigmatic systems gives a good introduction into characteristic typology and texture. However, their quantification offers a more objective way for the investigation of the considered system. With this quantification, the RPs have become more and more popular within a growing group of scientists who use RPs and their quantification techniques for data analysis (a search with the Scirus search engine in spring 2003 reveals over 200 journal published works and approximately 700 web published works about RPs).