{ "id": "ep3xq5cqw5", "slug": "recurrence-quantification-analysis", "title": "Recurrence quantification analysis", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Non-geographic · 1946 to 2000: Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" } ] } ], "excerpt": "Recurrence quantification analysis (RQA) is a nonlinear time-series method that quantifies a recurrence plot into measures such as determinism, laminarity, and entropy, introduced in the 1990s.", "snippet": "Recurrence quantification analysis (RQA) is a nonlinear time-series method that quantifies a recurrence plot into measures such as determinism, laminarity, and entropy, introduced in the 1990s.", "node": "physical.math.statistics", "markdown": "# Recurrence quantification analysis\n\nRecurrence quantification analysis (RQA) is a nonlinear time-series method that converts a [recurrence plot](https://www.edgepedia.org/recurrence-plot) into numbers, such as determinism, laminarity, and entropy, that characterize the dynamics underlying a measured series. The approaches based on recurrence plots are collectively referred to as RQA, and they quantify diagonal and vertical line structures that reflect repeated states and laminar phases.[1](https://www.mdpi.com/2571-631X/2/4/21) The method works with nominal, ordinal, or interval-scale data, makes no distributional assumptions, and is robust to outliers and non-stationarity.[2](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02232/full)\n\n| Key fact | Detail |\n|---|---|\n| Core object | The recurrence matrix \\( R_{i,j}(\\varepsilon) = \\Theta(\\varepsilon - \\|x_i - x_j\\|) \\), a Heaviside function of phase-space distances[3](https://doi.org/10.1016/j.physrep.2006.11.001) |\n| Main measures | Recurrence rate, determinism, laminarity, trapping time, divergence, and diagonal-line entropy[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf) • [5](https://tocsy.pik-potsdam.de/CRPtoolbox/crp_man.pdf) |\n| Determinism range | 0 to 1; a sinusoid gives 1, a purely stochastic signal a value extremely close to 0[1](https://www.mdpi.com/2571-631X/2/4/21) |\n| Length needed | About 50–100 data points for univariate RQA[1](https://www.mdpi.com/2571-631X/2/4/21); as few as 10–30 for cross-recurrence analysis[2](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02232/full) |\n| Threshold guidance | At least five times the noise standard deviation, \\( \\varepsilon > 5\\sigma \\)[3](https://doi.org/10.1016/j.physrep.2006.11.001); recurrence rate targets of 5–10%[1](https://www.mdpi.com/2571-631X/2/4/21) or 1–5% (up to 10%)[6](https://link.springer.com/article/10.1140/epjs/s11734-026-02338-3) are both advocated |\n| Computational cost | \\( O(N^{2}) \\) time and space for the \\( N \\times N \\) recurrence matrix[7](https://arxiv.org/pdf/2409.04110) |\n| Software | CRP Toolbox (MATLAB)[8](https://tocsy.pik-potsdam.de/CRPtoolbox/), crqa (R)[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf), PyRQA[7](https://arxiv.org/pdf/2409.04110), AccRQA[6](https://link.springer.com/article/10.1140/epjs/s11734-026-02338-3) |\n\n## How it works\n\nA recurrence plot is an array of dots in an \\( N \\times N \\) square, where a dot is placed at \\( (i, j) \\) whenever \\( x(j) \\) is sufficiently close to \\( x(i) \\).[9](https://iopscience.iop.org/article/10.1209/0295-5075/4/9/004/pdf) Formally, the scalar series is time-delay embedded into vectors \\( x_i \\) in an \\( m \\)-dimensional phase space, and the recurrence matrix is \\( R_{i,j}(\\varepsilon) = \\Theta(\\varepsilon - \\|x_i - x_j\\|) \\).[3](https://doi.org/10.1016/j.physrep.2006.11.001) Diagonal lines reflect repetitions of trajectories, while vertical lines mark states during which the series slows down, that is, laminar phases.[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf)\n\nThe standard measures are defined from the line-length histograms \\( P(l) \\) and \\( P(v) \\). Determinism is the fraction of recurrence points on diagonal lines of at least length \\( l_{\\min} \\): \\( \\mathrm{DET} = \\sum_{l=l_{\\min}}^{N} l\\,P(l) \\, / \\, \\sum_{l=1}^{N} l\\,P(l) \\).[3](https://doi.org/10.1016/j.physrep.2006.11.001) Laminarity is the analogous fraction for vertical lines, \\( \\mathrm{LAM} = \\sum_{v=v_{\\min}}^{N} v\\,P(v) \\, / \\, \\sum_{v=1}^{N} v\\,P(v) \\), and divergence is \\( \\mathrm{DIV} = 1/L_{\\max} \\), the inverse of the longest diagonal line, with the line of identity (and any chosen Theiler window) excluded so that it does not force \\( L_{\\max} \\) to equal the series length.[5](https://tocsy.pik-potsdam.de/CRPtoolbox/crp_man.pdf) The entropy of the diagonal-line distribution, \\( \\mathrm{ENTR} = -\\sum_{l=l_{\\min}}^{N} p(l) \\ln p(l) \\) with \\( p(l) = P(l)/N_l \\), measures the variety of line lengths.[3](https://doi.org/10.1016/j.physrep.2006.11.001) Trapping time (TT) is the average duration of a laminar phase.[10](https://complexity-methods.github.io/book/continous-rqa-measures.html) A high determinism with high diagonal-line entropy indicates a meta- or multi-stable regime, whereas high determinism with low entropy indicates a relatively stable regime.[10](https://complexity-methods.github.io/book/continous-rqa-measures.html) There is no specific guideline for choosing \\( l_{\\min} \\); \\( l_{\\min} = 1 \\) yields DET of 100%, and \\( l_{\\min} = 4 \\) is usually an appropriate choice.[11](https://www.sciencedirect.com/science/article/pii/S0895717710001986)\n\n## How it is done\n\nThe analysis proceeds in four parameter choices. The embedding dimension \\( m \\) is estimated with the false nearest neighbors algorithm, and the delay \\( \\tau \\) with the autocorrelation function, although the first minimum of the self-mutual information function is preferable.[1](https://www.mdpi.com/2571-631X/2/4/21) The threshold \\( \\varepsilon \\) is the hardest choice: rules of thumb include a few per cent of the maximum phase space diameter, not exceeding 10% of the mean or maximum diameter, keeping the recurrence rate near 1% of points,[3](https://doi.org/10.1016/j.physrep.2006.11.001) or setting \\( \\varepsilon \\) to the 5%-quantile of the distance distribution, which gives a 5% recurrence rate and robust characteristics across embedding dimensions.[7](https://arxiv.org/pdf/2409.04110) With observational noise of standard deviation \\( \\sigma \\), \\( \\varepsilon \\) should satisfy \\( \\varepsilon > 5\\sigma \\).[3](https://doi.org/10.1016/j.physrep.2006.11.001)\n\nRecurrence-based approaches perform reasonably well even when the series is short, about 50–100 data points,[1](https://www.mdpi.com/2571-631X/2/4/21) and cross-recurrence analyses can be used with as few as 10–30 points.[2](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02232/full) The recurrence matrix requires an \\( N \\times N \\) pairwise test, so costs are \\( O(N^{2}) \\), with quantification usually adding a further \\( O(N^{2}) \\).[7](https://arxiv.org/pdf/2409.04110) If the threshold is zero, the recurrence rate and diagonal-line measures such as determinism can be obtained in \\( O(N \\log(N)) \\) time and \\( O(N) \\) space, and approximations with the same reduced complexity exist for \\( \\varepsilon > 0 \\).[12](https://publications.pik-potsdam.de/rest/items/item_20541_2/component/file_20542/content) Approximative RQA computes series longer than 1 million points in seconds, where standard single-thread calculations need hours,[7](https://arxiv.org/pdf/2409.04110) and crqa's auto-RQA path scales to series with \\( N \\gg 10^{5} \\) with negligible memory footprint.[13](https://morenococo.r-universe.dev/crqa/doc/manual.html)\n\nSoftware implementations cover the main languages. The CRP Toolbox provides MATLAB routines for recurrence plots, phase-space reconstruction, extended RQA, recurrence networks, cross recurrence plots, and joint recurrence plots.[8](https://tocsy.pik-potsdam.de/CRPtoolbox/) The R package crqa unifies univariate, cross, windowed, and multivariate analysis, with semi-automatic parameter estimation (optimizeParam) and a piecewiseRQA function for long series.[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf) Python users have recurrence_python,[14](https://github.com/bmfreis/recurrence_python) PyRQA, and the AccRQA library.[6](https://link.springer.com/article/10.1140/epjs/s11734-026-02338-3) RecurrenceAnalysis.jl provides equivalent functions in Julia.[15](https://juliadynamics.github.io/RecurrenceAnalysis.jl/stable/quantification/)\n\n## Origin\n\nThe recurrence plot is a graphical tool for measuring the time constancy of dynamical systems, useful also when the assumptions needed for computing quantities such as information dimension, entropy, and Lyapunov exponents are not satisfied.[9](https://iopscience.iop.org/article/10.1209/0295-5075/4/9/004/pdf) RQA was developed to characterize the behavior of time series with nonlinear dynamics and is described as particularly versatile; published accounts credit its introduction to different years, 1992[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf) and 1994.[16](https://pure.mpg.de/rest/items/item_2425151_8/component/file_2505092/content) The field was consolidated in the review \"Recurrence plots for the analysis of complex systems\" by Norbert Marwan and colleagues in [Physics Reports](https://www.edgepedia.org/physics-reports) in 2007.[3](https://doi.org/10.1016/j.physrep.2006.11.001)\n\n## Variants\n\nCross-recurrence quantification analysis extends the univariate method to two time series, quantifying temporal coupling or similarity; the cross-recurrence matrix is \\( CR_{i,j} = \\Theta(\\varepsilon - \\|x_i - y_j\\|) \\), and unlike a recurrence plot it need not contain the line of identity.[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf) Further extensions include joint-recurrence plots for multivariate data and windowed-, meta-, weighted, fuzzy, and order-pattern recurrence plots.[1](https://www.mdpi.com/2571-631X/2/4/21) Order-pattern recurrence plots omit the need for selecting a threshold \\( \\varepsilon \\), which cannot be estimated easily.[17](http://www.dimigen.de/docs/Schinkel.et.al.2009.PhysLettA.pdf) For categorical series, an area-based entropy (catH) based on the distribution of rectangular structures in the plot provides more accurate entropy estimation than classic diagonal-line entropy.[4](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf) Averaged RQA computes measures over a range of thresholds (in fact recurrence rates) to omit the threshold choice, using a GPU algorithm and data with lengths from \\( 10^{2} \\) to \\( 10^{5} \\).[18](https://epjst.epj.org/articles/epjst/abs/2023/01/11734_2022_Article_686/11734_2022_Article_686.html) In 2025, Felipe Eduardo Lopes da Cruz and colleagues introduced recurrence microstate analysis (RMA) in [Physical Review E](https://www.edgepedia.org/physical-review-e), deriving determinism and laminarity from distributions of small \\( 3 \\times 3 \\) submatrices of the recurrence plot; the values are equivalent to the traditional line-based ones while sampling only about 5% of microstates, which avoids generating the full plot and enables real-time quantification.[19](https://doi.org/10.1103/physreve.111.044212)\n\n## Applications\n\nRecurrence plots have been applied in astrophysics, earth sciences, engineering, biology, cardiology, and neuroscience.[3](https://doi.org/10.1016/j.physrep.2006.11.001) CRQA, diagonal cross-recurrence profiles (DCRP), and multidimensional RQA (MdRQA) are widely used in the cognitive and social sciences for time-dependent behavioral and neurophysiological processes.[2](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02232/full) Recurrence plots have been combined with convolutional neural networks for classification and time-series prediction, and with machine learning for transition detection, monitoring, and anomaly detection.[7](https://arxiv.org/pdf/2409.04110) In gravitational physics, RQA applied to 22 simulated binary neutron star coalescences distinguished inspiral, merger, and post-merger stages with statistical validation at \\( p < 0.05 \\).[20](https://link.springer.com/article/10.1140/epjs/s11734-026-02538-x)\n\n## Limitations and alternatives\n\nHigh determinism is a necessary but not sufficient condition for determinism: the non-deterministic auto-regressive process \\( x_{i} = 0.8x_{i-1} + 0.3x_{i-2} - 0.25x_{i-3} + 0.9\\xi \\), with \\( \\xi \\) white Gaussian noise, yields DET of 0.6 at embedding dimension 4, delay 4, and fixed recurrence rate 0.1.[21](https://ar5iv.labs.arxiv.org/html/1007.2215) [Embedding](https://www.edgepedia.org/embedding) introduces spurious diagonal lines even in uncorrelated white noise, and low-pass filtering can also create spurious line structures, so a high DET alone must not be read as determinism.[21](https://ar5iv.labs.arxiv.org/html/1007.2215) Threshold selection is a key problem across disciplines[22](https://publications.pik-potsdam.de/rest/items/item_22560_5/component/file_24223/content) and a trade-off between a small threshold and a sufficient number of recurrences and recurrence structures.[21](https://ar5iv.labs.arxiv.org/html/1007.2215) The TREND measure is very sensitive to window size and can reveal contrary results, and detected nonstationarity in a finite series does not imply nonstationarity of the underlying system.[21](https://ar5iv.labs.arxiv.org/html/1007.2215) Standard practice provides no confidence bounds for DET or LAM, although a bootstrap method that resamples the line distributions \\( P(l) \\) and \\( P(v) \\) supplies confidence intervals at virtually no extra cost.[17](http://www.dimigen.de/docs/Schinkel.et.al.2009.PhysLettA.pdf) Interpreting \\( \\mathrm{DIV} = 1/L_{\\max} \\) as an estimator of the maximal [Lyapunov exponent](https://www.edgepedia.org/lyapunov-exponent) incorporates, in the words of that critique, \"high potential of erroneous conclusions derived from RQA\".[21](https://ar5iv.labs.arxiv.org/html/1007.2215) The short-series advantage holds only for the heuristic RQA measures; estimating dynamical invariants from recurrence plots still requires long time series.[21](https://ar5iv.labs.arxiv.org/html/1007.2215)\n\n## References\n\n1. [A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots](https://www.mdpi.com/2571-631X/2/4/21)\n2. [Analyzing Multivariate Dynamics Using CRQA, DCRP, and MdRQA – A Tutorial in R](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02232/full)\n3. [N MARWAN and colleagues (2007). Recurrence plots for the analysis of complex systems. Physics Reports.](https://doi.org/10.1016/j.physrep.2006.11.001)\n4. [Methods for Recurrence Quantification (crqa package, R Journal)](https://journal.r-project.org/articles/RJ-2021-062/RJ-2021-062.pdf)\n5. [Cross Recurrence Plot Toolbox manual](https://tocsy.pik-potsdam.de/CRPtoolbox/crp_man.pdf)\n6. [AccRQA library: accelerating recurrence quantification analysis (EPJ ST)](https://link.springer.com/article/10.1140/epjs/s11734-026-02338-3)\n7. [Recurrence plot review with computational-cost section (arXiv 2409.04110, 2024)](https://arxiv.org/pdf/2409.04110)\n8. [CRP Toolbox](https://tocsy.pik-potsdam.de/CRPtoolbox/)\n9. [Recurrence Plots of Dynamical Systems](https://iopscience.iop.org/article/10.1209/0295-5075/4/9/004/pdf)\n10. [Continuous RQA measures (The Complex Systems Approach to Behavioural Science)](https://complexity-methods.github.io/book/continous-rqa-measures.html)\n11. [Comparison of recurrence quantification methods for the analysis of temporal and spatial chaos](https://www.sciencedirect.com/science/article/pii/S0895717710001986)\n12. [Approximate recurrence quantification analysis (Spiegel et al.)](https://publications.pik-potsdam.de/rest/items/item_20541_2/component/file_20542/content)\n13. [Package 'crqa' reference manual](https://morenococo.r-universe.dev/crqa/doc/manual.html)\n14. [bmfreis/recurrence_python](https://github.com/bmfreis/recurrence_python)\n15. [Recurrence Quantification Analysis · RecurrenceAnalysis.jl](https://juliadynamics.github.io/RecurrenceAnalysis.jl/stable/quantification/)\n16. [MPG publication chapter citing Webber & Zbilut (1994) and Takens (1981)](https://pure.mpg.de/rest/items/item_2425151_8/component/file_2505092/content)\n17. [Confidence bounds of recurrence-based complexity measures (Physics Letters A, 2009)](http://www.dimigen.de/docs/Schinkel.et.al.2009.PhysLettA.pdf)\n18. [Averaged recurrence quantification analysis, method omitting the recurrence threshold choice (EPJ ST 232:47-56, 2023)](https://epjst.epj.org/articles/epjst/abs/2023/01/11734_2022_Article_686/11734_2022_Article_686.html)\n19. [Felipe Eduardo Lopes da Cruz and colleagues (2025). Density-based recurrence measures from microstates. Physical review. E.](https://doi.org/10.1103/physreve.111.044212)\n20. [Recurrence quantification analysis for binary neutron star coalescence simulations (EPJ ST)](https://link.springer.com/article/10.1140/epjs/s11734-026-02538-x)\n21. [How to avoid potential pitfalls in recurrence plot based data analysis](https://ar5iv.labs.arxiv.org/html/1007.2215)\n22. [On the appropriate selection of recurrence thresholds in RQA](https://publications.pik-potsdam.de/rest/items/item_22560_5/component/file_24223/content)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/recurrence-quantification-analysis", "markdown_url": "https://www.edgepedia.org/recurrence-quantification-analysis.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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Edgepedia Community License 1.0.", "speakable": "Recurrence quantification analysis is a nonlinear time-series method that quantifies a recurrence plot into measures such as determinism, laminarity, and entropy, introduced in the 1990s." }