Recurrence Plots and Cross Recurrence Plots

Recurrence Plots At A Glance


Recurrence plot – A recurrence plot (RP) is an advanced technique of nonlinear data analysis. It is a visualisation (or a graph) of a square matrix, in which the matrix elements correspond to those times at which a state of a dynamical system recurs (columns and rows correspond then to a certain pair of times). Techniqually, the RP reveals all the times when the phase space trajectory of the dynamical system visits roughly the same area in the phase space.

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André Sitz (AS-Internetdienst Potsdam), Norbert Marwan (Potsdam Institute for Climate Impact Research (PIK))
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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.

A Construction of recurrence plots B
(A) Segment of the phase space trajectory of the Lorenz system (for standard parameters \(r=28\), \(\sigma=10\), \(b=8/3\); Lorenz, 1963) by using its three components and (B) its corresponding distance matrix/ recurrence plot. A point of the trajectory at \(j\) which falls into the neighbourhood (gray circle in (A)) of a given point at \(i\) is considered as a recurrence point (black point on the trajectory in (A)). This is marked with a black point in the RP at the location \((i, j)\). A point outside the neighbourhood (small circle in (A)) causes a white point in the RP. The radius of the neighbourhood for the RP is \(\varepsilon=6\).

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.

Typology of recurrence plots
Characteristic typology of recurrence plots: (A) homogeneous (uniformly distributed noise), (B) periodic (super-positioned harmonic oscillations), (C) drift (logistic map corrupted with a linearly increasing term) and (D) disrupted (Brownian motion). These examples illustrate how different RPs can be. The used data have the length 400 (A, B, D) and 150 (C), respectively; no embeddings are used; the thresholds are \(\varepsilon=0.2\) (A, C, D) and \(\varepsilon=0.4\) (B).

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).

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).

» Recurrence plots in Wikipedia
» Further definitions of recurrence plots (Google)

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Please respect the copyrights! The content of this web site is protected by a Creative Commons License. You may use the text or figures, but you have to cite this source ( as well as N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007.

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