Prediction of flow dynamics using point processes

Yoshito  Hirata; Thomas  Stemler; Deniz  Eroglu; Norbert  Marwan

doi:10.1063/1.5016219

ABSTRACT

Describing a time series parsimoniously is the first step to study the underlying dynamics. For a time-discrete system, a generating partition provides a compact description such that a time series and a symbolic sequence are one-to-one. But, for a time-continuous system, such a compact description does not have a solid basis. Here, we propose to describe a time-continuous time series using a local cross section and the times when the orbit crosses the local cross section. We show that if such a series of crossing times and some past observations are given, we can predict the system's dynamics with fine accuracy. This reconstructability neither depends strongly on the size nor the placement of the local cross section if we have a sufficiently long database. We demonstrate the proposed method using the Lorenz model as well as the actual measurement of wind speed.

Y.H. would like to appreciate Professor Kazuyuki Aihara for the discussions. The research of Y.H. was supported by Kozo Keikaku Engineering, Inc. The research of D.E. has been supported by the German-Israeli Foundation for Scientific Research and Development (GIF), GIF Grant No. I-1298-415.13/2015.

REFERENCES

1. C. S. Greene, J. Tan, M. Ung, J. H. Moore, and C. Cheng, J. Cell. Physiol. 229, 1896–1900 (2014). https://doi.org/10.1002/jcp.24662, Google ScholarCrossref
2. W. Q. Meeker and Y. Hong, Qual. Eng. 26, 102–116 (2014). https://doi.org/10.1080/08982112.2014.846119, Google ScholarCrossref
3. M. McCullough, K. Sakellariou, T. Stemler, and M. Small, Chaos 26, 123103 (2016). https://doi.org/10.1063/1.4968551, Google ScholarScitation
4. K. Sakellariou, M. McCullough, T. Stemler, and M. Small, Chaos 26, 123104 (2016). https://doi.org/10.1063/1.4970483, Google ScholarScitation
5. Y. Hirata and K. Aihara, J. Neurosci. Methods 183, 277–286 (2009). https://doi.org/10.1016/j.jneumeth.2009.06.030, Google ScholarCrossref
6. Y. Hirata and K. Aihara, Physica A 391, 760–766 (2012). https://doi.org/10.1016/j.physa.2011.09.013, Google ScholarCrossref
7. Y. Hirata, E. J. Lang, and K. Aihara, “ Recurrence plots and the analysis of multiple spike trains,” in Springer Handbook of Bio-/Neuroinformatics, edited by N. Kasabov ( Springer, Berlin, Heidelberg, 2014), pp. 735–744. Google ScholarCrossref
8. I. Ozken, D. Eroglu, T. Stemler, N. Marwan, G. B. Bagci, and J. Kurths, Phys. Rev. E 91, 062911 (2015). https://doi.org/10.1103/PhysRevE.91.062911, Google ScholarCrossref
9. D. Eroglu, F. H. McRobie, I. Ozken, T. Stemler, K.-H. Wyrwoll, S. F. M. Breitenbach, N. Marwan, and J. Kurths, Nat. Commun. 7, 12929 (2016). https://doi.org/10.1038/ncomms12929, Google ScholarCrossref
10. J. D. Victor and K. P. Purpura, Network: Comput. Neural Syst. 8, 127–164 (1997). https://doi.org/10.1088/0954-898X_8_2_003, Google ScholarCrossref
11. H. Poincaré, “ Sur le problème des trois corps et les équations de la dynamique,” Acta Math. 13, 1–270 (1890). Google ScholarCrossref
12. N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, “ Recurrence plots for the analysis of complex systems,” Phys. Rep. 438(5–6), 237–329 (2007). https://doi.org/10.1016/j.physrep.2006.11.001, Google ScholarCrossref
13. Y. Hirata, K. Iwayama, and K. Aihara, Phys. Rev. E 94, 042217 (2016). https://doi.org/10.1103/PhysRevE.94.042217, Google ScholarCrossref
14. M. L. Van Quyen, J. Martinerie, C. Adam, and F. J. Varela, Physica D 127, 250–266 (1999). https://doi.org/10.1016/S0167-2789(98)00258-9, Google ScholarCrossref
15. E. N. Lorenz, J. Atmos. Sci. 20, 130–141 (1963). https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2, Google ScholarCrossref
16. F. Takens, Lect. Notes Math. 898, 366 (1981). https://doi.org/10.1007/BFb0091903, Google ScholarCrossref
17. T. Sauer, J. A. Yorke, and M. Casdagli, J. Stat. Phys. 65, 579 (1991). https://doi.org/10.1007/BF01053745, Google ScholarCrossref
18. T. Sauer, Phys. Rev. Lett. 72, 3811–3814 (1994). https://doi.org/10.1103/PhysRevLett.72.3811, Google ScholarCrossref
19. J. P. Huke and D. S. Broomhead, Nonlinearity 20, 2205–2244 (2007). https://doi.org/10.1088/0951-7715/20/9/011, Google ScholarCrossref
20. S. Smale, “ Stable manifolds for differential equations and diffeomorphisms,” in Topologia Differenziale ( Springer, 2011), pp. 93–126. Google ScholarCrossref
21. K. MacLeod, A. Bäcker, and G. Laurent, Nature 395, 693–698 (1998). https://doi.org/10.1038/27201, Google ScholarCrossref
22. R. Nomura, Y. Liang, and T. Okada, PLoS One 10, e0140774 (2015). https://doi.org/10.1371/journal.pone.0140774, Google ScholarCrossref
23. S. Suzuki, Y. Hirata, and K. Aihara, Int. J. Bifurcation Chaos 20, 3699–3708 (2010). https://doi.org/10.1142/S0218127410027970, Google ScholarCrossref
24. Y. Hirata and K. Aihara, Chaos 25, 123117 (2015). https://doi.org/10.1063/1.4938186, Google ScholarScitation
25. S. Shinomoto, K. Shima, and J. Tanji, Neural Comput. 15, 2823–2842 (2003). https://doi.org/10.1162/089976603322518759, Google ScholarCrossref
26. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. ( John Wiley & Sons, Hoboken, 2006). Google Scholar
27. Y. Hirata, D. P. Mandic, H. Suzuki, and K. Aihara, Philos. Trans. R. Soc. A 366, 591–607 (2008). https://doi.org/10.1098/rsta.2007.2112, Google ScholarCrossref
28. Y. Hirata, T. Takeuchi, S. Horai, H. Suzuki, and K. Aihara, Sci. Rep. 5, 15736 (2015). https://doi.org/10.1038/srep15736, Google ScholarCrossref
29. F. P. Schoenberg and K. E. Tranbarger, Environnmentrics 19, 271–286 (2008). https://doi.org/10.1002/env.867, Google ScholarCrossref
30. H. Hino, K. Takano, and N. Murata, R J. 7, 237–248 (2015). Google ScholarCrossref
31. K. Iwamaya, Y. Hirata, and K. Aihara, Phys. Lett. A 381, 257–262 (2017). https://doi.org/10.1016/j.physleta.2016.10.061, Google ScholarCrossref
32. M. B. Kennel, Phys. Rev. E 56, 316–321 (1997). https://doi.org/10.1103/PhysRevE.56.316, Google ScholarCrossref
33. P. Grassberger and H. Kantz, Phys. Lett. A 113, 235 (1985). https://doi.org/10.1016/0375-9601(85)90016-7, Google ScholarCrossref
34. R. L. Davidchack, Y.-C. Lai, E. M. Bollt, and M. Dhamala, Phys. Rev. E 61, 1353 (2000). https://doi.org/10.1103/PhysRevE.61.1353, Google ScholarCrossref
35. J. Plumecoq and M. Lefranc, Physica D 144, 231 (2000). https://doi.org/10.1016/S0167-2789(00)00082-8, Google ScholarCrossref
36. J. Plumecoq and M. Lefranc, Physica D 144, 259 (2000). https://doi.org/10.1016/S0167-2789(00)00083-X, Google ScholarCrossref
37. M. B. Kennel and M. Buhl, Phys. Rev. Lett. 91, 084102 (2003). https://doi.org/10.1103/PhysRevLett.91.084102, Google ScholarCrossref
38. Y. Hirata, K. Judd, and D. Kilminster, Phys. Rev. E 70, 016215 (2004). https://doi.org/10.1103/PhysRevE.70.016215, Google ScholarCrossref
39. P. König, A. K. Engel, and W. Singer, Trends Neurosci. 19, 130–137 (1996). https://doi.org/10.1016/S0166-2236(96)80019-1, Google ScholarCrossref
40. R. Azouz and C. M. Gray, Proc. Natl. Acad. Sci. U.S.A. 97, 8110–8115 (2000). https://doi.org/10.1073/pnas.130200797, Google ScholarCrossref
41. I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, IEEE Commun. Mag. 40, 102 (2002). https://doi.org/10.1109/MCOM.2002.1024422, Google ScholarCrossref
42. R. Liu, R. Shang, Y. Liu, and X. Lu, Remote Sens. Environ. 189, 164–179 (2017). https://doi.org/10.1016/j.rse.2016.11.023, Google ScholarCrossref
43. F. Clette, L. Svalgaard, J. M. Vaquero, and E. W. Cliver, Space Sci. Rev. 186, 35–103 (2014). https://doi.org/10.1007/s11214-014-0074-2, Google ScholarCrossref
44. Y. Aono and K. Kazui, Int. J. Climatol. 28, 905–914 (2008). https://doi.org/10.1002/joc.1594, Google ScholarCrossref

Prediction of flow dynamics using point processes

REFERENCES

Article Metrics

Altmetric

Resources

General Information

1.8K

REFERENCES

Article Metrics

Altmetric