Recurrence plot: Difference between revisions

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Eckmann et al. (1987) introduced recurrence plots, which provide a way to visualize the periodic nature of a trajectory through a [[phase space]]. Often, the phase space does not have a low enough dimension (two or three) to be pictured, since higher-dimensional phase spaces can only be visualized by projection into the two or three-dimensional sub-spaces. However, making a recurrence plot enables us to investigate certain aspects of the ''m''-dimensional phase space trajectory through a two-dimensional representation.

A '''recurrence''' is a time the trajectory returns to a ___location it has visited before. The recurrence plot depicts the collection of pairs of times at which the trajectory is at the same place, i.e. the set of <math>(i,j)</math> with <math>\vec{x}(i) = \vec{x}(j)</math>. This can show many things: for instance, if the trajectory is strictly periodic with period <math>T</math>, then all such pairs of times will be separated by a multiple of <math>T</math> and visible as diagonal lines. To make the plot, continuous time and continuous phase space are discretized, taking e.g. <math>\vec{x}(i)</math> as the ___location of the trajectory at time <math>i \tau</math> and counting as a recurrence any time the trajectory gets sufficiently close (say, within &epsilon;) to a point it has been previously. Concretely then, recurrence/non-recurrence can be recorded by the binary function

Operationally the plot is drawn as follows:

(a) A certain time window <math>\vec{w} = <t_0, t_1, ...t, ..., t_T></math> is chosen where any two time steps are separated by the time interval <math>\varepsilon</math>, and where the state <math>\vec{x}(t) </math> of the system is recorded for each time step, thus collecting the trajectory <math> \mathbf{X} = <\vec{x}(t_0), \vec{x}(t_1), ..., \vec{x}(t_T)> </math>.

(b) A 2D plot is created where the x-axis and y-axis both report <math>\vec{w}</math>, forming a <math>T \times T</math> lattice of little squares each with side measuring <math>\varepsilon</math>

(c) The data <math>\mathbf{X}</math> are used to compute a matrix <math>\mathbf{D}_{T,T}</math> formed by binary elements recording the recurrence/non-recurrence of values <math>\vec{x}</math> through the binary function:

:<math>R(i,j) = \begin{cases} 1 &\text{if} \quad \| \vec{x}(i) - \vec{x}(j)\| \le \varepsilon \\ 0 & \text{otherwise}, \end{cases}</math>

where <math>i,j \in \{t_0, t_1, ...t, ..., t_T\}</math>.

and the recurrence plot puts a (black) point at coordinates <math>(i,j)</math> if <math>R(i,j)=1</math>.

(d) The recurrence plot then visualises <math>\mathbf{D}_{T,T}</math> with a black little square of the lattice at coordinates <math>(i,j)</math> if <math>R(i,j)=1</math>, and a white little square if <math>R(i,j)=0</math>.

The visual appearance of a recurrence plot gives hints about the dynamics of the system. 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''. TheFor visualexample, appearancethe plot can show if the trajectory is strictly periodic with period <math>T</math>, then all such pairs of antimes RPwill givesbe hintsseparated aboutby thea dynamicsmultiple of the<math>T</math> systemand visible as diagonal lines.

[[Image:Rp examples740.gif|thumb|center|740px|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 [[autoregressive process|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 invariant]]s 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 <math>y</math>-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.