Recurrence plot: Difference between revisions

Revision as of 14:36, 6 September 2022

Revision as of 14:42, 6 September 2022

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==Detailed description==

One way to visualize the recurring nature of astates by their 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 has before).

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:<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 \| \cdot \|</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>, with time at the <math>x</math>- and <math>y</math>-axes.

If only a [[time series]] is available, the phase space can be reconstructed by using a time delay embedding (see [[Takens' theorem]]):

:<math>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),</math>

where <math>u(i)</math> is the time series, <math>m</math> the embedding dimension and <math>\tau</math> the time delay.

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.