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Detecting dynamical changes in time series by using the Jensen Shannon divergence. (English) Zbl 1390.37134
Summary: Most of the time series in nature are a mixture of signals with deterministic and random dynamics. Thus the distinction between these two characteristics becomes important. Distinguishing between chaotic and aleatory signals is difficult because they have a common wide band power spectrum, a delta like autocorrelation function, and share other features as well. In general, signals are presented as continuous records and require to be discretized for being analyzed. In this work, we introduce different schemes for discretizing and for detecting dynamical changes in time series. One of the main motivations is to detect transitions between the chaotic and random regime. The tools here used here originate from the Information Theory. The schemes proposed are applied to simulated and real life signals, showing in all cases a high proficiency for detecting changes in the dynamics of the associated time series.
©2017 American Institute of Physics
37M10 Time series analysis of dynamical systems
Full Text: DOI
[1] Wold, H., A Study in the Analysis of Stationary Time Series, (1938), Almqvist & Wiksell · JFM 64.1200.02
[2] Cencini, M.; Falcioni, M.; Olbrich, E.; Kantz, H.; Vulpiani, A., Chaos or noise: Difficulties of a distinction, Phys. Rev. E, 62, 1, 427, (2000)
[3] Sakai, H.; Tokumaru, H., Autocorrelations of a certain chaos, IEEE Trans. Acoust. Speech Signal Process., 28, 5, 588-590, (1980) · Zbl 0525.93055
[4] Rosso, O. A.; Larrondo, H. A.; Martin, M. T.; Plastino, A.; Fuentes, M. A., Distinguishing noise from chaos, Phys. Rev. Lett., 99, 15, 154102, (2007)
[5] Zunino, L.; Soriano, M. C.; Rosso, O. A., Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach, Phys. Rev. E, 86, 4, 046210, (2012)
[6] Olivares, F.; Plastino, A.; Rosso, O. A., Contrasting chaos with noise via local versus global information quantifiers, Phys. Lett. A, 376, 19, 1577-1583, (2012) · Zbl 1260.37022
[7] Brock, W. A., Distinguishing random and deterministic systems: Abridged version, J. Econ. Theory, 40, 1, 168-195, (1986) · Zbl 0616.62125
[8] Gao, J. B.; Hu, J.; Tung, W. W.; Cao, Y. H., Distinguishing chaos from noise by scale-dependent Lyapunov exponent, Phys. Rev. E, 74, 6, 066204, (2006)
[9] Elsner, J. B., Predicting time series using a neural network as a method of distinguishing chaos from noise, J. Phys. A: Math. General, 25, 4, 843, (1992)
[10] Wu, Z.; Huang, N. E., A study of the characteristics of white noise using the empirical mode decomposition method, Proc. R. Soc. London A: Math. Phys. Eng. Sci., 460, 1597-1611, (2004) · Zbl 1062.62005
[11] Cover, T. M.; Thomas, J. A., Elements of Information Theory, (2012), John Wiley & Sons
[12] Mischaikow, K.; Mrozek, M.; Reiss, J.; Szymczak, A., Construction of symbolic dynamics from experimental time series, Phys. Rev. Lett., 82, 6, 1144, (1999)
[13] Powell, G. E.; Percival, I. C., A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems, J. Phys. A, 12, 11, 2053, (1979)
[14] Rosso, O. A.; Craig, H.; Moscato, P., Shakespeare and other english renaissance authors as characterized by information theory complexity quantifiers, Phys. A: Stat. Mech. Appl., 388, 6, 916-926, (2009)
[15] Rosso, O. A.; Blanco, S.; Yordanova, J.; Kolev, V.; Figliola, A.; Schürmann, M.; Başar, E., Wavelet entropy: A new tool for analysis of short duration brain electrical signals, J. Neurosci. Methods, 105, 1, 65-75, (2001)
[16] Bandt, C.; Pompe, B., Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88, 174102, (2002)
[17] Yang, A. C.; Hseu, S.; Yien, H.; Goldberger, A. L.; Peng, C., Linguistic analysis of the human heartbeat using frequency and rank order statistics, Phys. Rev. Lett., 90, 10, 108103, (2003)
[18] Burbera, J.; Rao, C. R., Entropy, differential metric, distance and divergence measures in probability spaces: a unified approach, J. Multivariate Statist., 12, 537-596, (1982) · Zbl 0526.60015
[19] Lin, J., Divergence measures based on the shannon entropy, Inf. Theory, IEEE Trans., 37, 1, 145-151, (1991) · Zbl 0712.94004
[20] López-Rosa, S.; Antolín, J.; Angulo, J. C.; Esquivel, R. O., Divergence analysis of atomic ionization processes and isoelectronic series, Phys. Rev. A, 80, 1, 012505, (2009)
[21] Angulo, J. C.; Antolín, J.; López-Rosa, S.; Esquivel, R. O., Jensen-Shannon divergence in conjugate spaces: The entropy excess of atomic systems and sets with respect to their constituents, Phys. A: Stat. Mech. Appl., 389, 4, 899-907, (2010)
[22] Antolín, J.; Angulo, J. C.; López-Rosa, S., Fisher and jensen-shannon divergences: Quantitative comparisons among distributions: Application to position and momentum atomic densities, J. Chem. Phys., 130, 7, 074110, (2009)
[23] Martín, A. L.; López-Rosa, S.; Angulo, J. C.; Antolín, J., Jensen-Shannon and Kullback-Leibler divergences as quantifiers of relativistic effects in neutral atoms, Chem. Phys. Lett., 635, 75-79, (2015)
[24] Grosse, I.; Bernaola-Galván, P.; Carpena, P.; Román-Roldán, R.; Oliver, J.; Stanley, H. E., Analysis of symbolic sequences using the Jensen-Shannon divergence, Phys. Rev. E, 65, 4, 041905, (2002) · Zbl 1245.94057
[25] Endres, D. M.; Schindelin, J. E., A new metric for probability distributions, IEEE Trans. Inf. Theory, 49, 7, 1858-1860, (2003) · Zbl 1294.62003
[26] Nguyen, H.-V.; Vreeken, J., Non-parametric Jensen-Shannon divergence, Joint European Conference on Machine Learning and Knowledge Discovery in Databases, 173-189, (2015), Springer
[27] Beirlant, J.; Dudewicz, E. J.; Györfi, L.; Van der Meulen, E. C., Nonparametric entropy estimation: An overview, Int. J. Math. Stat. Sci., 6, 1, 17-39, (1997) · Zbl 0882.62003
[28] Robinson, J. C., Dimensions, Embeddings, and Attractors, 186, (2011), Cambridge University Press
[29] Takens, F., Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick 1980, 366-381, (1981), Springer · Zbl 0513.58032
[30] Zozor, S.; Mateos, D.; Lamberti, P. W., Mixing bandt-pompe and lempel-ziv approaches: Another way to analyze the complexity of continuous-state sequences, Eur. Phys. J. B, 87, 5, 107, (2014)
[31] Sprott, J. C., Chaos and Time-Series Analysis, 69, (2003), Oxford University Press: Oxford University Press, Oxford
[32] Knuth, D. E., The Art of Computer Programming, (1998), Adison-Wesley · Zbl 0895.68054
[33] Steeb, W. H.; Van Wy, M. A., Chaos and Fractals: Algorithms and Computations, (1992), Wissenschaftsverlag
[34] May, R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 5560, 459-467, (1976) · Zbl 1369.37088
[35] Potapov, A.; Ali, M. K., Robust chaos in neural networks, Phys. Lett. A, 277, 6, 310-322, (2000) · Zbl 1167.82359
[36] Ricker, W. E., Stock and recruitment, J. Fish. Board Can., 11, 5, 559-623, (1954)
[37] Arnol’d, V. I., Small denominators. I. Mapping the circle onto itself, Izv. Ross. Akad. Nauk. Ser. Mat., 25, 1, 21-86, (1961)
[38] Strogatz, S. H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology and Chemistry, (2001), Perseus Publishing
[39] Shaw, R., Chaotic Behavior and Information Flow, Z. Naturforsch., 38a, 80-112, (1981) · Zbl 0599.58033
[40] Hirsch, M. W.; Smale, S.; Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, (2004), Academic Press · Zbl 1135.37002
[41] Hénon, M., A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50, 1, 69-77, (1976) · Zbl 0576.58018
[42] Aronson, D. G.; Chory, M. A.; Hall, G. R.; McGehee, R. P., Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Commun. Math. Phys., 83, 3, 303-354, (1982) · Zbl 0499.70034
[43] Nusse, H. E., Dynamics: Numerical Explorations, (1998), Springer
[44] Schmidt, G.; Wang, B. W., Dissipative standard map, Phys. Rev. A, 32, 5, 2994, (1985)
[45] Arnold, V. I.; Avez, A., Ergodic Problems of Classical Mechanics, (1968), Benjamin · Zbl 0167.22901
[46] Chernikov, A. A.; Sagdeev, R. Z.; Zaslavskii, G. M., Chaos-how regular can it be?, Phys. Today, 41, 27-35, (1988)
[47] Chirikov, B. V., A universal instability of many-dimensional oscillator systems, Phys. Rep., 52, 5, 263-379, (1979)
[48] Devaney, R. L., A piecewise linear model for the zones of instability of an area-preserving map, Phys. D: Nonlinear Phenom., 10, 3, 387-393, (1984) · Zbl 0588.58009
[49] Larrondo, H. A., (2012)
[50] Rabin, M. O., Probabilistic algorithm for testing primality, J. Number Theory, 12, 1, 128-138, (1980) · Zbl 0426.10006
[51] Is the normal heart rate chaotic, (2008)
[52] Blödt, M., Granjon, P., Raison, B., and Regnier, J., “ Mechanical fault detection in induction motor drives through stator current monitoring-theory and application examples,” Fault Detect. W. Zhang (Ed), 451-488 (2010).
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