Recurrence plot: Difference between revisions

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One way to visualize the recurring nature of a trajectory through a [[phase space]] is the recurrence plot, introduced by Eckmann et al. (1987). 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.

At a '''recurrence''' the trajectory returns to a ___location in phase space it has visited before up to a small error <math>\varepsilon</math> (i.e., the system returns to a state that it hadhas before).
The recurrence plot represents the collection of pairs of times such recurrences, i.e., the set of <math>(i,j)</math> with <math>\vec{x}(i) \approx \vec{x}(j)</math>, with <math>i</math> and <math>j</math> discrete points of time and <math>\vec{x}(i</math> the state of the system at time <math>i</math> (___location of the trajectory at time <math>i</math>).

Mathematically, this can be expressed by the binary recurrence matrix

Operationally the plot is drawn as follows:

(a) A certain time window <math>\vec{w} = \left \langle t_1, t_2, ..., t_T \right \rangle</math> is chosen where any two successive time steps are separated by the time interval <math>\delta</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} = \left\langle \vec{x}(t_1), \vec{x}(t_2), ..., \vec{x}(t_T)\right\rangle </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>\delta</math>

(c) The data <math>\mathbf{X}</math> are used to compute a matrix <math>\mathbf{R}</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> \quad \| \cdots \|</math> is a norm and <math>\varepsilon</math> the recurrence threshold. The recurrence plot visualises <math>\mathbf{R}</math> with coloured (mostly black) dot at coordinates <math>(i,j)</math> if <math>R(i,j)=1</math>.

where <math>i,j \in \{t_1, t_2, ..., t_T\}</math>.

(d) The recurrence plot then visualises <math>\mathbf{R}</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''. For example, the plot can show 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.