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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>\| \cdot \|</math> is a norm and <math>\varepsilon</math> the recurrence threshold. An alternative, more formal expression is using the [[Heaviside step function]] <math>R(i,j)=\Theta(\varepsilon - D_{i,j})</math>
with <math>D_{i,j} = \| \vec{x}(i)- \vec{x}(j) \|</math> the norm of distance vector between <math>\vec{x}(i)</math> and <math>\vec{x}(j)</math>.
Alternative recurrence definitions consider different distances <math>D_{i,j}</math>, e.g., [[angular distance]], [[fuzzy set|fuzzy distance]], or [[Levenshtein distance|edit distance]]<ref name="marwan2023">{{cite journal
| author1=N. Marwan | author2=K. H. Kraemer
| title=Trends in recurrence analysis of dynamical systems
| journal=European Physical Journal ST
| volume=232
| year=2023
| pages=5–27
| doi=10.1140/epjs/s11734-022-00739-8
|bibcode = 2023EPJST.232....5M
| s2cid=255630484
}}</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.
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[[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 ([[logistic map]]) with linear trend, and data from an [[autoregressive process|auto-regressive process]].]]
The small-scale structures in recurrence plots contain information about certain characteristics of the dynamics of the underlying system. For example, the length of the diagonal lines visible in the recurrence plot are related to the divergence of phase space trajectories, thus, can represent information about the chaoticity<ref name="marwan2007">{{cite journal
|author1=N. Marwan |author2=M. C. Romano |author3=M. Thiel |author4=J. Kurths | title=Recurrence Plots for the Analysis of Complex Systems
| journal=Physics Reports
Line 54 ⟶ 65:
| doi=10.1016/j.physrep.2006.11.001
| pages=237
|bibcode = 2007PhR...438..237M }}</ref>. Therefore, the [[recurrence quantification analysis]] quantifies the distribution of these small-scale structures<ref name="marwan2008">{{cite journal
|bibcode = 2007PhR...438..237M }}</ref>. This quantification allows us 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.▼
| author=N. Marwan
| title=A historical review of recurrence plots
| journal=European Physical Journal ST
| volume=164
| issue=1
| year=2008
| pages=3–12
| url= https://zenodo.org/record/996840
| doi=10.1140/epjst/e2008-00829-1
|bibcode = 2008EPJST.164....3M | arxiv=1709.09971
| s2cid=119494395
▲
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)<ref name="marwan2008"/>.
The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail.
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