Recurrence plot: Difference between revisions

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{{Short description|Type of plot in descriptive statistics and chaos theory}}

In descriptive [[statistics]] and [[chaos theory]], a '''recurrence plot''' ('''RP''') is a plot showing, for each moment <math>j</math> in time, the times at which the state of a [[dynamical system]] returns to the previous state at <math>i</math>,

i.e., when the [[phase space]] trajectory visits roughly the same area in the phase space as at time <math>j</math>. In other words, it is a plot of

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| bibcode=1987EL......4..973E

| s2cid=250847435

}}</ref> 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. One frequently used tool to study the behaviour of such phase space trajectories is then the [[Poincaré map]]. Another tool, is the recurrence plot, which enables us to investigate many aspects of the ''m''-dimensional phase space trajectory through a two-dimensional representation.

At a '''recurrence''' the trajectory returns to a ___location (state) in phase space it has visited before up to a small error <math>\varepsilon</math> . The recurrence plot represents the collection of pairs of times of 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>).

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| volume=232

| year=2023

| issue=1
| pages=5–27

| doi=10.1140/epjs/s11734-022-00739-8

|bibcode = 2023EPJST.232....5M

| s2cid=255630484

| doi-access=free

| arxiv=2409.04110

}}</ref>

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 univariate [[time series]] <math>u(it)</math> is available, the phase space can be reconstructed, e.g., 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 (with <math>t = i \Delta t</math> and <math>\Delta t</math> the sampling time), <math>m</math> the embedding dimension and <math>\tau</math> the time delay. PhaseHowever, phase space reconstruction is not essential part of the recurrence plot (although often stated in literature), because it is based on phase space trajectories which could be derived from the system's variables directly (e.g., from the three variables of the [[Lorenz system]]) or from multivariate data.

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.

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| doi=10.1016/j.physrep.2006.11.001

| pages=237

|bibcode = 2007PhR...438..237M | arxiv=2501.13933}}</ref> Therefore, the [[recurrence quantification analysis]] quantifies the distribution of these small-scale structures.<ref>{{cite journal

| author1=J. P. Zbilut | author2=C. L. Webber

| title=Embeddings and delays as derived from quantification of recurrence plots

| journal=Physics Letters A

| volume=171

| issue=3-43–4

| year=1992

| pages=199–203

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| issue=2

| year=1994

| pages=965-973965–973

| doi=10.1152/jappl.1994.76.2.965

| s2cidpmid=238545408175612

| s2cid=23854540

}}</ref><ref name="marwan2008">{{cite journal

| author=N. Marwan

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| year=2002

| doi=10.1016/S0375-9601(02)01170-2

| pages=299-307299–307

|bibcode = 2002PhLA..302..299M

| s2cid=8020903 | arxiv=physics/0201061

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| journal=Physics Letters A

| volume=336

| issue=4-54–5

| year=2005

| doi=10.1016/j.physleta.2004.12.056

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| journal=Physics Letters A

| volume=330

| issue=3-43–4

| year=2004

| doi=10.1016/j.physleta.2004.07.066