# Recurrence plot

﻿
Recurrence plot

In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space. In other words, it is a graph of

$\vec{x}(i)\approx \vec{x}(j),\,$

showing i on a horizontal axis and j on a vertical axis, where $\vec{x}$ is a phase space trajectory.

## Background

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 again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf. Poincaré recurrence theorem). The recurrence of states in nature has been known for a long time and has also been discussed in early work (e.g. Henri Poincaré 1890).

## Detailed description

Eckmann et al. (1987) introduced recurrence plots, which can visualize the recurrence of states in a phase space. Usually, a phase space does not have a low enough dimension (two or three) 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 and a different time j is pictured within a two-dimensional squared matrix with black and white dots, where black dots mark a recurrence, and both axes are time axes. This representation is called recurrence plot. Such a recurrence plot can be mathematically expressed as

$\mathbf{R}(i,j) = H(\varepsilon - || \vec{x}(i) - \vec{x}(j)||), \quad \vec{x}(i) \in \Bbb{R}^m, \quad i, j=1, \dots, N,$

where N is the number of considered states $\vec{x}(i)$, ε is a threshold distance, || · || a norm (e.g. Euclidean norm) and H the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):

$\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),$

where u(i) is the time series, m the embedding dimension and τ the time delay.

Caused by characteristic behaviour of the phase space trajectory, a recurrence plot contains typical small-scale structures, as single dots, diagonal lines and vertical/horizontal lines (or a mixture of the latter, which combines to extended clusters). The large-scale structure, also called texture, can be visually characterised by homogenous, periodic, drift or disrupted. The visual appearance of an RP gives hints about the dynamics of the system.

Typical examples of recurrence plots (top row: time series (plotted over time); bottom row: corresponding recurrence plots). From left to right: uncorrelated stochastic data (white noise), harmonic oscillation with two frequencies, chaotic data with linear trend (logistic map) and data from an auto-regressive process.

The small-scale structures in RPs are used by the recurrence quantification analysis (Zbilut & Webber 1992; Marwan et al. 2002). This quantification allows to describe the RPs in a quantitative way, and to study transitions or nonlinear parameters of the system. In contrast to the heuristic approach of the recurrence quantification analysis, which depends on the choice of the embedding parameters, some dynamical invariants as correlation dimension, K2 entropy or mutual information, which are independent on the embedding, can also be derived from recurrence plots. The base for these dynamical invariants are the recurrence rate and the distribution of the lengths of the diagonal lines.

Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the y-axis (instead of absolute time).

The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail.

## Extensions

Multivariate extensions of recurrence plots were developed as cross recurrence plots and joint recurrence plots.

Cross recurrence plots consider the phase space trajectories of two different systems in the same phase space (Marwan & Kurths 2002):

$\mathbf{CR}(i,j) = \Theta(\varepsilon - || \vec{x}(i) - \vec{y}(j)||), \quad \vec{x}(i),\, \vec{y}(i) \in \Bbb{R}^m, \quad i=1, \dots, N_x, \ j=1, \dots, N_y.$

The dimension of both systems must be the same, but the number of considered states (i.e. data length) can be different. Cross recurrence plots compare the simultaneous occurrences of similar states of two systems. They can be used in order to analyse the similarity of the dynamical evolution between two different systems, to look for similar matching patterns in two systems, or to study the time-relationship of two similar systems, whose time-scale differ (Marwan & Kurths 2005).

Joint recurrence plots are the Hadamard product of the recurrence plots of the considered sub-systems (Romano et al. 2004), e.g. for two systems $\vec{x}$ and $\vec{y}$ the joint recurrence plot is

$\mathbf{JR}(i,j) = \Theta(\varepsilon_x - || \vec{x}(i) - \vec{x}(j)||) \cdot \Theta(\varepsilon_y - || \vec{y}(i) - \vec{y}(j)||), \quad \vec{x}(i) \in \Bbb{R}^m, \quad \vec{y}(i) \in \Bbb{R}^n,\quad i,j=1, \dots, N_{x,y}.$

In contrast to cross recurrence plots, joint recurrence plots compare the simultaneous occurrence of recurrences in two (or more) systems. Moreover, the dimension of the considered phase spaces can be different, but the number of the considered states has to be the same for all the sub-systems. Joint recurrence plots can be used in order to detect phase synchronisation.

## Example

Recurrence plot of the Southern Oscillation index.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Recurrence plot — (engl: recurrence Wiederkehr) bedeutet Wiederkehr Darstellung und ist eine moderne Methode der nichtlinearen Datenanalyse. Die Wiederkehr Eigenschaft ist typisch für deterministische dynamische Systeme (Chaos, nichtlineare Dynamik) und spiegelt… …   Deutsch Wikipedia

• Recurrence quantification analysis — (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory.BackgroundThe… …   Wikipedia

• Recurrence period density entropy — (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal. Overview Recurrence period density entropy is useful for characterising the… …   Wikipedia

• Recurrence — and recurrent may refer to: *Recurrence relation, an equation which defines a sequence recursively *Poincaré recurrence theorem, Henri Poincaré s theorem on dynamical systems *Recurrence plot, a statistical plot that shows a pattern that re… …   Wikipedia

• Plot (graphics) — Scatterplot of the eruption interval for Old Faithful (a geyser). A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a… …   Wikipedia

• Close returns plot — Recurrence Plot (engl: recurrence Wiederkehr) bedeutet Wiederkehr Darstellung und ist eine moderne Methode der nichtlinearen Datenanalyse. Die Wiederkehr Eigenschaft ist typisch für deterministische dynamische Systeme (Chaos, nichtlineare… …   Deutsch Wikipedia

• Dot plot (bioinformatics) — This article is about the biological sequences comparison plot. For the statistical plot, see Dot plot (statistics). A dot plot (a.k.a. contact plot or residue contact map) is a graphical method that allows the comparison of two biological… …   Wikipedia

• Poincaré recurrence theorem — In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence.… …   Wikipedia

• Dot-Plot — Ein DNA Dotplot des menschlichen Zink Finger Transkriptionsfaktors (GenBank ID NM 002383) zeigt regionale Selbst Ähnlichkeiten. Ein Dotplot (dt. Punktauftragung) ist eine graphische Methode der Bioinformatik zwei biologische Sequenzen miteinander …   Deutsch Wikipedia

• Dot plot — Ein DNA Dotplot des menschlichen Zink Finger Transkriptionsfaktors (GenBank ID NM 002383) zeigt regionale Selbst Ähnlichkeiten. Ein Dotplot (dt. Punktauftragung) ist eine graphische Methode der Bioinformatik zwei biologische Sequenzen miteinander …   Deutsch Wikipedia