imported>OAbot: Open access bot: arxiv updated in citation with #oabot.

2025-02-03T03:15:18Z

Open access bot: arxiv updated in citation with #oabot.

New page

'''Recurrence quantification analysis''' ('''RQA''') is a method of [[nonlinear]] [[data analysis]] (cf. [[chaos theory]]) for the investigation of [[dynamical systems]]. It quantifies the number and duration of recurrences of a dynamical system presented by its [[phase space]] trajectory.<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
| volume=438
| issue=5–6
| year=2007
| doi=10.1016/j.physrep.2006.11.001
| pages=237
|bibcode = 2007PhR...438..237M | arxiv=2501.13933}}</ref>

==Background==
The recurrence quantification analysis (RQA) was developed in order to quantify differently appearing [[recurrence plot]]s (RPs), based on the small-scale structures therein.<ref name="marwan2008">{{cite journal
| 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
}}</ref> [[Recurrence plot]]s are tools which visualise the recurrence behaviour of the [[phase space]] trajectory <math>\vec{x}(i)</math> of [[dynamical systems]]:<ref>{{cite journal
| author=J. P. Eckmann, S. O. Kamphorst, [[David Ruelle|D. Ruelle]]
| title=Recurrence Plots of Dynamical Systems
| journal=Europhysics Letters
| volume=5
| issue=9
| pages=973–977
| year=1987
| doi=10.1209/0295-5075/4/9/004
| bibcode=1987EL......4..973E
| s2cid=250847435
}}</ref>

:<math>{R}(i,j) = \Theta(\varepsilon - \| \vec{x}(i) - \vec{x}(j)\|)</math>,

where <math>\Theta: \mathbf{R} \rightarrow \{0, 1\}</math> is the [[Heaviside function]] and <math>\varepsilon</math> a predefined tolerance.

Recurrence plots mostly contain single dots and lines which are parallel to the mean diagonal (''line of identity'', LOI) or which are vertical/horizontal. Lines parallel to the LOI are referred to as ''diagonal lines'' and the vertical structures as ''vertical lines''. Because an RP is usually symmetric, horizontal and vertical lines correspond to each other, and, hence, only vertical lines are considered. The lines correspond to a typical behaviour of the phase space trajectory: whereas the diagonal lines represent such segments of the phase space trajectory which run parallel for some time, the vertical lines represent segments which remain in the same [[phase space]] region for some time.<ref name="marwan2007"></ref>

If only a univariate [[time series]] <math>u(t)</math> 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 (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. However, pPhase space reconstruction is not essential part of the RQA (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 RQA quantifies the small-scale structures of recurrence plots, which present the number and duration of the recurrences of a dynamical system. The measures introduced for the RQA were developed heuristically between 1992 and 2002.<ref name="zbilut1992">{{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=3–4
| year=1992
| pages=199–203
| doi=10.1016/0375-9601(92)90426-M
| bibcode = 1992PhLA..171..199Z
| s2cid=122890777
}}</ref><ref name="webber1994">{{cite journal
| author1=C. L. Webber | author2=J. P. Zbilut
| title=Dynamical assessment of physiological systems and states using recurrence plot strategies
| journal=Journal of Applied Physiology
| volume=76
| issue=2
| year=1994
| pages=965–973
| doi=10.1152/jappl.1994.76.2.965
| pmid=8175612
| s2cid=23854540
}}</ref><ref name="marwan2002">{{cite journal
| author1=N. Marwan | author2=N. Wessel | author3=U. Meyerfeldt | author4=A. Schirdewan | author5=J. Kurths
| title=Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data
| journal=Physical Review E
| volume=66
| issue=2
| year=2002
| pages=026702
| url= https://arxiv.org/pdf/physics/0201064
| doi=10.1103/PhysRevE.66.026702
| arxiv = physics/0201064
| bibcode = 2002PhRvE..66b6702M
| pmid=12241313
| s2cid=14803032
}}</ref> They are actually [[Complexity|measures of complexity]]. The main advantage of the RQA is that it can provide useful information even for short and non-stationary data, where other methods fail.

RQA can be applied to almost every kind of data. It is widely used in [[physiology]], but was also successfully applied on problems from [[engineering]], [[chemistry]], [[Earth sciences]] etc.<ref name="marwan2008"></ref>
Further extensions and variations of measures for quantifying recurrence properties have been proposed to address specific research questions. RQA measures are also combined with machine learning approaches for classification tasks.<ref name="marwan2023">{{cite journal
| author1=N. Marwan | author2=K. H. Kraemer
| title=Trends in recurrence analysis of dynamical systems
| journal=European Physical Journal – Special Topics
| volume=232
| year=2023
| pages=5–27
| url=https://arxiv.org/pdf/2409.04110
| doi=10.1140/epjs/s11734-022-00739-8
| doi-access=free
| arxiv = 2409.04110
| bibcode = 2023EPJST.232....5M
| s2cid=255630484
}}</ref>

==RQA measures==
The simplest measure is the '''recurrence rate''', which is the density of recurrence points in a recurrence plot:<ref name="marwan2007"></ref>

:<math>\text{RR} = \frac{1}{N^2} \sum_{i,j=1}^N {R}(i,j).</math>

The recurrence rate corresponds with the probability that a specific state will recur. It is almost equal with the definition of the [[correlation sum]], where the LOI is excluded from the computation.

The next measure is the percentage of recurrence points which form diagonal lines in the recurrence plot of minimal length <math>\ell_\min</math>:<ref name="webber1994"></ref>

:<math>\text{DET} = \frac{\sum_{\ell=\ell_\min}^N \ell\, P(\ell)}{\sum_{\ell=1}^{N}\ell P(\ell)},</math>

where <math>P(\ell)</math> is the [[frequency distribution]] of the lengths <math>\ell</math> of the diagonal lines (i.e., it counts how many instances have length <math>\ell</math>). This measure is called '''determinism''' and is related with the [[predictability]] of the [[dynamical system]], because [[white noise]] has a recurrence plot with almost only single dots and very few diagonal lines, whereas a [[deterministic process]] has a recurrence plot with very few single dots but many long diagonal lines.

The number of recurrence points which form vertical lines can be quantified in the same way:<ref name="marwan2002"></ref>

:<math> \text{LAM} = \frac{\sum_{v=v_\min}^{N}vP(v)}{\sum_{v=1}^{N}vP(v)},</math>

where <math>P(v)</math> is the frequency distribution of the lengths <math>v</math> of the vertical lines, which have at least a length of <math>v_\min</math>. This measure is called '''laminarity''' and is related with the amount of [[laminar phase]]s in the system ([[intermittency]]).

The lengths of the diagonal and vertical lines can be measured as well. The '''averaged diagonal line length'''<ref name="webber1994"></ref>

:<math>\text{L} = \frac{\sum_{\ell=\ell_\min}^N \ell\, P(\ell)}{\sum_{\ell=\ell_\min}^N P(\ell)}</math>

is related with the ''predictability time'' of the dynamical system
and the '''trapping time''', measuring the average length
of the vertical lines,<ref name="marwan2002"></ref>

:<math>\text{TT} = \frac{\sum_{v=v_\min}^{N} v P(v)} {\sum_{v=v_\min}^{N} P(v)}</math>

is related with the ''laminarity time'' of the dynamical system, i.e. how long the system remains in a specific state.<ref name="marwan2002"></ref>

Because the length of the diagonal lines is related on the time how long segments of the [[phase space]] trajectory run parallel, i.e. on the [[divergence]] behaviour of the trajectories, it was sometimes stated that the [[Multiplicative inverse|reciprocal]] of the maximal length of the diagonal lines (without LOI) would be an estimator for the positive maximal [[Lyapunov exponent]] of the dynamical system. Therefore, the '''maximal diagonal line length''' <math>L_\max</math> or the '''divergence''':<ref name="marwan2007"></ref>

:<math>\text{DIV} = \frac{1}{L_\max}</math>

are also measures of the RQA. However, the relationship between these measures with the positive maximal Lyapunov exponent is not as easy as stated, but even more complex (to calculate the Lyapunov exponent from an RP, the whole frequency distribution of the diagonal lines has to be considered). The divergence can have the trend of the positive maximal Lyapunov exponent, but not more. Moreover, also RPs of white noise processes can have a really long diagonal line, although very seldom, just by a finite probability. Therefore, the divergence cannot reflect the maximal Lyapunov exponent.

The [[probability]] <math>p(\ell)</math> that a diagonal line has exactly length <math>\ell</math> can be estimated from the frequency distribution <math>P(\ell)</math> with <math>p(\ell) = \frac{P(\ell)}{\sum_{\ell=l_\min}^N P(\ell)}</math>. The [[Shannon entropy]] of this probability,<ref name="webber1994"></ref>

:<math>\text{ENTR} = - \sum_{\ell=\ell_\min}^N p(\ell) \ln p(\ell),</math>

reflects the complexity of the deterministic structure in the system. However, this entropy depends sensitively on the bin number and, thus, may differ for different realisations of the same process, as well as for different data preparations.

The last measure of the RQA quantifies the thinning-out of the recurrence plot. The '''trend''' is the regression coefficient of a linear relationship between the density of recurrence points in a line parallel to the LOI and its distance to the LOI. More exactly, consider the recurrence rate in a diagonal line parallel to LOI of distance ''k'' (''diagonal-wise recurrence rate'' or ''τ-recurrence rate''):<ref name="marwan2007"></ref>

:<math>\text{RR}_k = \frac{1}{N-k} \sum_{j-i=k}^{N-k} {R}(i,j),</math>

then the trend is defined by<ref name="webber1994"></ref>

:<math>\text{TREND} = \frac{\sum_{i=1}^\tilde{N} (i-\tilde{N}/2)(RR_i - \langle RR_i \rangle)}{\sum_{i=1}^\tilde{N} (i-\tilde{N}/2)^2},</math>

with <math>\langle \cdot \rangle</math> as the average value and <math>\tilde{N} < N</math>. This latter relation should ensure to avoid the edge effects of too low recurrence point densities in the edges of the recurrence plot. The measure ''trend'' provides information about the stationarity of the system.

Similar to the <math>\tau</math>-recurrence rate, the other measures based on the diagonal lines (DET, L, ENTR) can be defined diagonal-wise. These definitions are useful to study interrelations or [[synchronisation]] between different systems (using [[recurrence plot]]s or [[recurrence plot#Extensions|cross recurrence plots]]).<ref>{{Cite journal| author=Marwan, N., Kurths, J. | year=2002 | title=Nonlinear analysis of bivariate data with cross recurrence plots | journal=Physics Letters A | volume=302 | pages=299–307 | doi=10.1016/S0375-9601(02)01170-2|arxiv = physics/0201061 |bibcode = 2002PhLA..302..299M| issue=5–6 | s2cid=8020903 }}</ref>

==Time-dependent RQA==
Instead of computing the RQA measures of the entire recurrence plot, they can be computed in small windows moving over the recurrence plot along the LOI. This provides time-dependent RQA measures which allow detecting, e.g., chaos-chaos transitions.<ref>{{Cite journal| author1=L. L. Trulla | author2=A. Giuliani | author3=J. P. Zbilut | author4=C. L. Webber, Jr. | year=1996 | title=Recurrence quantification analysis of the logistic equation with transients | journal=Physics Letters A | volume=223 | issue=4 | pages=255–260 | doi=10.1016/S0375-9601(96)00741-4 }}</ref><ref name="marwan2007"></ref> '''Note:''' the choice of the size of the window can strongly influence the measure ''trend''.

==Example==
[[Image:LogisticMap BifurcationDiagram.png|thumb|left|512px|Bifurcation diagram for the Logistic map.]]

[[Image:Logistic map rqa.svg|thumb|left|820px|RQA measures of the logistic map for various setting of the control parameter a. The measures RR and DET exhibit maxima at chaos-order/ order-chaos transitions. The measure DIV has a similar trend as the maximal [[Lyapunov exponent]] (but it is not the same!). The measure LAM has maxima at chaos-chaos transitions ([[laminar phase]]s, [[intermittency]]).]]
{{Clear}}

==See also==
* [[Recurrence plot]], a powerful visualisation tool of recurrences in dynamical (and other) systems.
* [[Recurrence period density entropy]], an information-theoretic method for summarising the recurrence properties of both deterministic and stochastic dynamical systems.
*[[Approximate entropy]]

==References==
{{reflist}}

==External links==
* http://www.recurrence-plot.tk/

{{DEFAULTSORT:Recurrence Quantification Analysis}}
[[Category:Signal processing]]
[[Category:Dynamical systems]]
[[Category:Chaos theory]]
[[Category:Nonlinear time series analysis]]

Recurrence quantification analysis - Revision history

imported>OAbot: Open access bot: arxiv updated in citation with #oabot.