Skip to main content
Springer Nature Link
Log in

How to compute suitable vicinity parameter and sampling time of recurrence analysis

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We show how the maximum of recurrence-microstates entropy configures a new way to properly compute appropriate recurrence vicinity parameter and time sampling to perform recurrence analysis of continuous data. For experimental data, we show the same procedure may be used to find the optimum sampling or to perform down-sampling of the data, preserving recurrence meaning and adjusting the ideal sampling. The new method retrieves results obtained using traditional methods with the advantage of being independent of any free parameter, since all parameter dependencies are automatically set. Our results are also less sensitive to noise when experimental data are used. Due to the automatized way to capture suitable recurrence parameters, the method is adequate for using in autonomous numerical algorithms, allowing the recovery of relevant recurrence information embedded in time series (including over-sampled data), rationalizing the process of data acquisition and allowing only relevant data to be collected.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. The SGAMP database containing single-unit neuronal recordings of North Atlantic squid (Loligo pealei) giant axons in response to stochastic stimulus currents is provided by PhysioNet [10, 23]. The data are open and free available at https://physionet.org/content/sgamp/1.0.0/.

References

  1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer USA, New York (1996). https://doi.org/10.1007/0-387-22492-0_3

  2. Chawla, N.V., Bowyer, K.W., Hall, L.O., Kegelmeyer, W.P.: Smote: synthetic minority over-sampling technique. J. Artif. Intell. Res. 16, 321–357 (2002)

    Article  Google Scholar 

  3. Chua, L.O., Wu, C.W., Huang, A.: Guo-Qun Zhong: a universal circuit for studying and generating chaos—part I: routes to chaos. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40(10), 732–744 (1993)

    Article  Google Scholar 

  4. Corso, G., Prado, T.D.L., dos Santos Lima, G.Z., Kurths, J., Lopes, S.R.: Quantifying entropy using recurrence matrix microstates. Chaos Interdiscip. J. Nonlinear Sci. 28(8), 083108 (2018). https://doi.org/10.1063/1.5042026

    Article  MathSciNet  Google Scholar 

  5. Eckmann, J.P., Kamphorst, S.O., Ruelle, D.: Recurrence plots of dynamical systems. Europhys. Lett. 4(9), 973–977 (1987). https://doi.org/10.1209/0295-5075/4/9/004

    Article  Google Scholar 

  6. Eroglu, D., Marwan, N., Prasad, S., Kurths, J.: Finding recurrence networks’ threshold adaptively for a specific time series. Nonlinear Process. Geophys. 21(6), 1085–1092 (2014)

    Article  Google Scholar 

  7. Frank, R.J., Davey, N., Hunt, S.P.: Time series prediction and neural networks. J. Intell. Robot. Syst. 31(1), 91–103 (2001). https://doi.org/10.1023/A:1012074215150

    Article  Google Scholar 

  8. Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140 (1986). https://doi.org/10.1103/PhysRevA.33.1134

    Article  MathSciNet  Google Scholar 

  9. Froguel, L.B., de Lima Prado, T., Corso, G., dos Santos Lima, G.Z., Lopes, S.R.: Efficient computation of recurrence quantification analysis via microstates. Appl. Math. Comput. 428, 127175 (2022). https://doi.org/10.1016/j.amc.2022.127175

    Article  MathSciNet  Google Scholar 

  10. Goldberger, A.L., Amaral, L.A., Glass, L., Hausdorff, J.M., Ivanov, P.C., Mark, R.G., Mietus, J.E., Moody, G.B., Peng, C.K., Stanley, H.E.: Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation [Online] 101(23), e215–e220 (2000)

    Google Scholar 

  11. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957). https://doi.org/10.1103/PhysRev.106.620

    Article  MathSciNet  Google Scholar 

  12. Kantz, H., Schreiber, T.: Nonlinear time series analysis, 2 edn. Cambridge University Press (2004)

  13. Kapur, J.N., Kesavan, H.K.: The generalized maximum entropy principle (with applications). Sandford Educational Press Canada (1987)

  14. Kasdin, N.J.: Discrete simulation of colored noise and stochastic processes and 1/f/sup/spl alpha//power law noise generation. Proc. IEEE 83(5), 802–827 (1995)

    Article  Google Scholar 

  15. Kraskov, A., Stögbauer, H., Grassberger, P.: Estimating mutual information. Phys. Rev. E 69, 066138 (2004). https://doi.org/10.1103/PhysRevE.69.066138

    Article  MathSciNet  Google Scholar 

  16. Kwapień, J., Drożdż, S., Liu, L., Ioannides, A.: Cooperative dynamics in auditory brain response. Phys. Rev. E 58, 6359–6367 (1998). https://doi.org/10.1103/PhysRevE.58.6359

    Article  Google Scholar 

  17. Lopes, S., Prado, T., Corso, G., dos Santos Lima, G.Z., Kurths, J.: Parameter-free quantification of stochastic and chaotic signals. Chaos Solitons Fractals 133, 109616 (2020). https://doi.org/10.1016/j.chaos.2020.109616

    Article  MathSciNet  Google Scholar 

  18. Marwan, N.: How to avoid potential pitfalls in recurrence plot based data analysis. Int. J. Bifurc. Chaos 21(04), 1003–1017 (2011)

    Article  MathSciNet  Google Scholar 

  19. Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Phys. Rep. 438(5), 237–329 (2007)

    Article  MathSciNet  Google Scholar 

  20. Mewett, D.T., Reynolds, K.J., Nazeran, H.: Recurrence plot features: An example using ecg. In: ISSPA’99. Proceedings of the Fifth International Symposium on Signal Processing and its Applications (IEEE Cat. No. 99EX359), vol. 1, pp. 175–178. IEEE (1999)

  21. Parlitz, U.: Lyapunov exponents from chua’s circuit. J. Circuits Syst. Comput. 3(02), 507–523 (1993)

    Article  Google Scholar 

  22. Parlitz, U.: Estimating Lyapunov Exponents from Time Series, pp. 1–34. Springer Berlin Heidelberg, Berlin, Heidelberg (2016). https://doi.org/10.1007/978-3-662-48410-4_1

  23. Paydarfar, D., Forger, D.B., Clay, J.R.: Noisy inputs and the induction of on-off switching behavior in a neuronal pacemaker. J. Neurophysiol. 96(6), 3338–3348 (2006). https://doi.org/10.1152/jn.00486.2006. (PMID: 16956993)

    Article  Google Scholar 

  24. Prado, T.L., Boaretto, B.R.R., Corso, G., dos Santos Lima, G.Z., Kurths, J., Lopes, S.R.: A direct method to detect deterministic and stochastic properties of data. New J. Phys. 24(3), 033027 (2022). https://doi.org/10.1088/1367-2630/ac5057

    Article  MathSciNet  Google Scholar 

  25. Prado, T.L., Corso, G., dos Santos Lima, G.Z., Budzinski, R.C., Boaretto, B.R.R., Ferrari, F.A.S., Macau, E.E.N., Lopes, S.R.: Maximum entropy principle in recurrence plot analysis on stochastic and chaotic systems. Chaos Interdiscip. J. Nonlinear Sci. 30(4), 043123 (2020). https://doi.org/10.1063/1.5125921

    Article  MathSciNet  Google Scholar 

  26. Priestley, M.B.: Non-linear and non-stationary time series analysis. Academic Press (1988)

  27. Pyle, D.: Data preparation for data mining. Morgan kaufmann (1999)

  28. Renyi, A.: Probability Theory. North-Holland, Amsterdam (1970)

  29. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest lyapunov exponents from small data sets. Phys. D Nonlinear Phenom. 65(1), 117–134 (1993). https://doi.org/10.1016/0167-2789(93)90009-P

    Article  MathSciNet  Google Scholar 

  30. Schinkel, S., Dimigen, O., Marwan, N.: Selection of recurrence threshold for signal detection. Eur. Phys. J. Special Top. 164(1), 45–53 (2008)

    Article  Google Scholar 

  31. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Techn. J. 27(3), 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

    Article  MathSciNet  Google Scholar 

  32. Thiel, M., Romano, M.C., Kurths, J.: Analytical description of recurrence plots of white noise and chaotic processes (2003). https://doi.org/10.48550/ARXIV.NLIN/0301027

  33. Thiel, M., Romano, M.C., Kurths, J., Meucci, R., Allaria, E., Arecchi, F.T.: Influence of observational noise on the recurrence quantification analysis. Phys. D Nonlinear Phenom. 171(3), 138–152 (2002)

    Article  MathSciNet  Google Scholar 

  34. von Toussaint, U.: Bayesian inference in physics. Rev. Mod. Phys. 83, 943–999 (2011). https://doi.org/10.1103/RevModPhys.83.943

    Article  Google Scholar 

  35. Webber, C.L., Zbilut, J.P.: Dynamical assessment of physiological systems and states using recurrence plot strategies. J. Appl. Physiol. 76(2), 965–973 (1994)

    Article  Google Scholar 

  36. Yang, D., Ren, W.X., Hu, Y.D., Li, D.: Selection of optimal threshold to construct recurrence plot for structural operational vibration measurements. J. Sound Vib. 349, 361–374 (2015)

    Article  Google Scholar 

  37. Zayed, A.I.: Advances in Shannon’s sampling theory. Routledge (2018)

  38. Zbilut, J.P., Webber, C.L.: Embeddings and delays as derived from quantification of recurrence plots. Phys. Lett. A 171(3–4), 199–203 (1992)

    Article  Google Scholar 

  39. Zbilut, J.P., Zaldivar-Comenges, J.M., Strozzi, F.: Recurrence quantification based liapunov exponents for monitoring divergence in experimental data. Phys. Lett. A 297(3–4), 173–181 (2002)

    Article  Google Scholar 

Download references

Funding

We thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq—Brazil, grant numbers 308441/2021-4, 307907/2019-8, and 308621/2019-0 and Financiadora de Estudos e Projetos (FINEP) for financial support (LCPAD, CtInfra project).

Author information

Authors and Affiliations

  1. Departamento de Física, Universidade Federal do Paraná, Curitiba, PR, 81531-980, Brazil

    Thiago de Lima Prado, Vandertone Santos Machado & Sergio Roberto Lopes

  2. Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil

    Gilberto Corso

  3. Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil

    Gustavo Zampier dos Santos Lima

Authors
  1. Thiago de Lima Prado
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Vandertone Santos Machado
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Gilberto Corso
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Gustavo Zampier dos Santos Lima
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Sergio Roberto Lopes
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

TLP was involved in the conceptualization; data curation; formal analysis; investigation; methodology; project administration; validation; and visualization. VSN contributed to the investigation and supporting. GC assisted in the investigation and supporting. GZSL contributed to the investigation and supporting. SRL contributed to the conceptualization; formal analysis; investigation; methodology; supervision; writing—original draft; writing—review & editing.

Corresponding author

Correspondence to Sergio Roberto Lopes.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Lima Prado, T., Machado, V.S., Corso, G. et al. How to compute suitable vicinity parameter and sampling time of recurrence analysis. Nonlinear Dyn 112, 1141–1152 (2024). https://doi.org/10.1007/s11071-023-09063-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09063-9

Keywords

Profiles

  1. Thiago de Lima Prado View author profile