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The reliability of recurrence network analysis is influenced by the observability properties of the recorded time series. (English) Zbl 1420.37112

Summary: Recurrence network analysis (RNA) is a remarkable technique for the detection of dynamical transitions in experimental applications. However, in practical experiments, often only a scalar time series is recorded. This requires the state-space reconstruction from this single time series which, as established by embedding and observability theory, is shown to be hampered if the recorded variable conveys poor observability. In this work, we investigate how RNA metrics are impacted by the observability properties of the recorded time series. Following the framework of Y. Zou et al. [“Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods”, Chaos 20, 043130, (2010; doi:10.1063/1.3523304)], we use the Rössler and Duffing-Ueda systems as benchmark models for our study. It is shown that usually RNA metrics perform badly with variables of poor observability as for recurrence quantification analysis. An exception is the clustering coefficient, which is rather robust to observability issues. Along with its efficacy to detect dynamical transitions, it is shown to be an efficient tool for RNA – especially when no prior information of the variable observability is available.
©2019 American Institute of Physics

MSC:

37M10 Time series analysis of dynamical systems
37N35 Dynamical systems in control
93B07 Observability

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K2
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[1] Kalman, R., On the general theory of control systems, IRE Trans. Automat. Contr., 4, 110-110 (1959) · doi:10.1109/TAC.1959.1104873
[2] Aguirre, L. A.; Portes, L. L.; Letellier, C., Structural, dynamical and symbolic observability: From dynamical systems to networks, PLoS One, 13, e0206180 (2018) · doi:10.1371/journal.pone.0206180
[3] Webber, Jr., J.; Zbilut, C. L., Dynamical assessment of physiological systems and states using recurrence plot strategies, J. Appl. Physiol., 76, 965-973 (1994) · doi:10.1152/jappl.1994.76.2.965
[4] Trulla, L.; Giuliani, A.; Zbilut, J.; Webber, C., Recurrence quantification analysis of the logistic equation with transients, Phys. Lett. A, 223, 255-260 (1996) · Zbl 1037.37507 · doi:10.1016/S0375-9601(96)00741-4
[5] Marwan, N.; Donges, J. F.; Zou, Y.; Donner, R. V.; Kurths, J., Complex network approach for recurrence analysis of time series, Phys. Lett. A, 373, 4246-4254 (2009) · Zbl 1234.05214 · doi:10.1016/j.physleta.2009.09.042
[6] Eckmann, J.-P.; Kamphorst, S. O.; Ruelle, D., Recurrence plots of dynamical systems, Europhys. Lett., 4, 973-977 (1987) · doi:10.1209/0295-5075/4/9/004
[7] Marwan, N.; Carmen Romano, M.; Thiel, M.; Kurths, J., Recurrence plots for the analysis of complex systems, Phys. Rep., 438, 237-329 (2007) · doi:10.1016/j.physrep.2006.11.001
[8] Carvalho, N. C.; Portes, L. L.; Beda, A.; Tallarico, L. M. S.; Aguirre, L. A., Recurrence plots for the assessment of patient-ventilator interactions quality during invasive mechanical ventilation, Chaos, 28, 085707 (2018) · doi:10.1063/1.5020371
[9] Zhang, J.; Small, M., Complex network from pseudoperiodic time series: Topology versus dynamics, Phys. Rev. Lett., 96, 238701 (2006) · doi:10.1103/PhysRevLett.96.238701
[10] Watts, D. J.; Strogatz, S. H., Collective dynamics of “small-world“ networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139 · doi:10.1038/30918
[11] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: Structure and dynamics, Phys. Rep., 424, 175-308 (2006) · Zbl 1371.82002 · doi:10.1016/j.physrep.2005.10.009
[12] Donner, R. V.; Small, M.; Donges, J. F.; Marwan, N.; Zou, Y.; Xiang, R.; Kurths, J., Recurrence-based time series analysis by means of complex network methods, Int. J. Bifurc. Chaos, 21, 1019-1046 (2011) · Zbl 1247.37086 · doi:10.1142/S0218127411029021
[13] Donner, R. V.; Zou, Y.; Donges, J. F.; Marwan, N.; Kurths, J., Recurrence networks—A novel paradigm for nonlinear time series analysis, New J. Phys., 12, 033025 (2010) · Zbl 1360.90045 · doi:10.1088/1367-2630/12/3/033025
[14] Zou, Y.; Donner, R. V.; Donges, J. F.; Marwan, N.; Kurths, J. J., Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods, Chaos, 20, 043130-043130 (2010) · doi:10.1063/1.3523304
[15] Gallas, J. A., Structure of the parameter space of the Hénon map, Phys. Rev. Lett., 70, 2714-2717 (1993) · doi:10.1103/PhysRevLett.70.2714
[16] Gallas, J. A. C., Dissecting shrimps: Results for some one-dimensional physical models, Phys. A Stat. Mech. Appl., 202, 196-223 (1994) · doi:10.1016/0378-4371(94)90174-0
[17] Stoop, R.; Martignoli, S.; Benner, P.; Stoop, R. L.; Uwate, Y., Shrimps: Occurrence, scaling and relevance, Int. J. Bifurc. Chaos, 22, 1230032 (2012) · Zbl 1258.37056 · doi:10.1142/S0218127412300327
[18] Bonatto, C.; Garreau, J. C.; Gallas, J. A. C., Self-similarities in the frequency-amplitude space of a loss-modulated CO \({}_2\) laser, Phys. Rev. Lett., 95, 2-5 (2005) · doi:10.1103/PhysRevLett.95.143905
[19] Oliveira, D. F.; Robnik, M.; Leonel, E. D., Shrimp-shape domains in a dissipative kicked rotator, Chaos, 21, 043122 (2011) · doi:10.1063/1.3657917
[20] Lekscha, J.; Donner, R. V., Phase space reconstruction for non-uniformly sampled noisy time series, Chaos, 28, 085702 (2018) · Zbl 1396.86011 · doi:10.1063/1.5023860
[21] Xiang, R.; Zhang, J.; Xu, X. K.; Small, M., Multiscale characterization of recurrence-based phase space networks constructed from time series, Chaos, 22, 013107 (2012) · doi:10.1063/1.3673789
[22] Eroglu, D.; Marwan, N.; Prasad, S.; Kurths, J., Finding recurrence networks’ threshold adaptively for a specific time series, Nonlinear Process Geophys., 21, 1085-1092 (2014) · doi:10.5194/npg-21-1085-2014
[23] Aguirre, L. A.; Bastos, S. B.; Alves, M. A.; Letellier, C., Observability of nonlinear dynamics: Normalized results and a time-series approach, Chaos, 18, 013123 (2008) · doi:10.1063/1.2885386
[24] Letellier, C.; Aguirre, L. A., Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables, Chaos, 12, 549-558 (2002) · Zbl 1080.37600 · doi:10.1063/1.1487570
[25] Aguirre, L. A.; Letellier, C., Observability of multivariate differential embeddings, J. Phys. A Math. Gen., 38, 6311-6326 (2005) · Zbl 1330.37078 · doi:10.1088/0305-4470/38/28/004
[26] Letellier, C.; Aguirre, L. A.; Maquet, J., Relation between observability and differential embeddings for nonlinear dynamics, Phys. Rev. E, 71, 066213 (2005) · doi:10.1103/PhysRevE.71.066213
[27] Aguirre, L. A.; Letellier, C., Investigating observability properties from data in nonlinear dynamics, Phys. Rev. E, 83, 066209 (2011) · doi:10.1103/PhysRevE.83.066209
[28] Portes, L. L.; Benda, R. N.; Ugrinowitsch, H.; Aguirre, L. A., Impact of the recorded variable on recurrence quantification analysis of flows, Phys. Lett. A, 378, 2382-2388 (2014) · Zbl 1303.37034 · doi:10.1016/j.physleta.2014.06.014
[29] Portes, L. L.; Aguirre, L. A., Enhancing multivariate singular spectrum analysis for phase synchronization: The role of observability, Chaos, 26, 093112 (2016) · doi:10.1063/1.4963013
[30] Aguirre, L. A.; Portes, L. L.; Letellier, C., Observability and synchronization of neuron models, Chaos, 27, 103103 (2017) · Zbl 1390.93164 · doi:10.1063/1.4985291
[31] Hermann, R.; Krener, A., Nonlinear controllability and observability, IEEE Trans. Automat. Contr., 22, 728-740 (1977) · Zbl 0396.93015 · doi:10.1109/TAC.1977.1101601
[32] Isidori, A., Nonlinear Control Systems, Communications and Control Engineering (Springer, London, 1995), p. 549. · Zbl 0878.93001
[33] Friedland, B., Controllability index based on conditioning number, J. Dyn. Syst. Meas. Control, 97, 444-445 (1975) · Zbl 0322.93005 · doi:10.1115/1.3426963
[34] Aguirre, L. A., Controllability and observability of linear systems: Some noninvariant aspects, IEEE Trans. Educ., 38, 33-39 (1995) · doi:10.1109/13.350218
[35] Letellier, C.; Maquet, J.; Sceller, L. L.; Gouesbet, G.; Aguirre, L. A., On the non-equivalence of observables in phase-space reconstructions from recorded time series, J. Phys. A Math. Gen., 31, 7913-7927 (1998) · Zbl 0936.81014 · doi:10.1088/0305-4470/31/39/008
[36] Donner, R. V.; Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J., The geometry of chaotic dynamics—A complex network perspective, Eur. Phys. J. B, 84, 653-672 (2011) · Zbl 1515.37092 · doi:10.1140/epjb/e2011-10899-1
[37] Zou, Y.; Donner, R. V.; Marwan, N.; Donges, J. F.; Kurths, J., Complex network approaches to nonlinear time series analysis, Phys. Rep., 787, 1-97 (2019) · doi:10.1016/j.physrep.2018.10.005
[38] Costa, L. D. F.; Rodrigues, F. A.; Travieso, G.; Villas Boas, P. R., Characterization of complex networks: A survey of measurements, Adv. Phys., 56, 167-242 (2007) · doi:10.1080/00018730601170527
[39] Newman, M. E. J., Assortative mixing in networks, Phys. Rev. Lett., 89, 208701 (2002) · doi:10.1103/PhysRevLett.89.208701
[40] Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 397-398 (1976) · Zbl 1371.37062 · doi:10.1016/0375-9601(76)90101-8
[41] Sandri, M., Numerical calculation of Lyapunov exponents, Math. J., 6, 78-84 (1996)
[42] Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.-M., Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15, 9-20 (1980) · Zbl 0488.70015 · doi:10.1007/BF02128236
[43] Jacob, R.; Harikrishnan, K. P.; Misra, R.; Ambika, G., Uniform framework for the recurrence-network analysis of chaotic time series, Phys. Rev. E, 93, 012202 (2016) · doi:10.1103/PhysRevE.93.012202
[44] Ueda, Y., Random phenomena resulting from non-linearity in the system described by Duffing’s equation, Int. J. Non-Linear Mech., 20, 481-491 (1985) · doi:10.1016/0020-7462(85)90024-1
[45] Bonatto, C.; Gallas, J. A. C.; Ueda, Y., Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator, Phys. Rev. E, 77, 026217 (2008) · doi:10.1103/PhysRevE.77.026217
[46] Maes, F.; Collignon, A.; Vandermeulen, D.; Marchal, G.; Suetens, P., Multimodality image registration by maximization of mutual information, IEEE Trans. Med. Imaging, 16, 187-198 (1997) · doi:10.1109/42.563664
[47] Kullback, S.; Leibler, R. A., On information and sufficiency, Ann. Math. Stat., 22, 79-86 (1951) · Zbl 0042.38403 · doi:10.1214/aoms/1177729694
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