Detection of Dynamical Complexity Changes in Dst Time Series Using Entropy Concepts and Rescaled Range Analysis

  • Georgios Balasis
  • Ioannis A. Daglis
  • Anastasios Anastasiadis
  • Konstantinos Eftaxias
Part of the IAGA Special Sopron Book Series book series (IAGA, volume 3)


Using an array of diagnostic tools including entropy concepts and rescaled range analysis, we establish that the Dst index time series exhibits long-range correlations, and that the underlying stochastic process can be modeled as fractional Brownian motion. We show the emergence of two distinct patterns in the geomagnetic variability of the terrestrial magnetosphere: (1) a pattern associated with intense magnetic storms, which is characterized by a higher degree of organization (i.e., lower complexity or higher predictability for the system) and persistent behavior, and (2) a pattern associated with normal periods, which is characterized by a lower degree of organization (i.e., higher complexity or lower predictability for the system) and anti-persistent behavior.


Magnetic Storm Shannon Entropy Fractional Brownian Motion Hurst Exponent Entropy Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The Dst data are provided by the World Data Center for Geomagnetism, Kyoto (


  1. Balasis G, Eftaxias K (2009) A study of non-extensivity in the Earth’s magnetosphere. Eur Phys J Special Topics 174:219–225CrossRefGoogle Scholar
  2. Balasis G, Daglis IA, Kapiris P, Mandea M, Vassiliadis D, Eftaxias K (2006) From prestorm activity to magnetic storms: a transition described in terms of fractal dynamics. Ann Geophys 24:3557–3567CrossRefGoogle Scholar
  3. Balasis G, Daglis IA, Papadimitriou C, Kalimeri M, Anastasiadis A, Eftaxias K (2008) Dynamical complexity in Dst time series using non-extensive Tsallis entropy. Geophys Res Lett. doi:10.1029/2008GL034743Google Scholar
  4. Balasis G, Daglis IA, Papadimitriou C, Kalimeri M, Anastasiadis A, Eftaxias K (2009) Investigating dynamical complexity in the magnetosphere using various entropy measures. J Geophys Res. doi:10.1029/2008JA014035Google Scholar
  5. Carbone A, Stanley H (2007) Scaling properties and entropy of long-range correlated time series. Physica A 384:267–271CrossRefGoogle Scholar
  6. Daglis IA, Baker DN, Galperin Y, Kappenman JG, Lanzerotti LJ (2001) Technological impacts of space storms: outstanding issues. Eos Trans AGU. doi:10.1029/01EO00340Google Scholar
  7. Daglis IA, Kozyra J, Kamide Y, Vassiliadis D, Sharma A, Liemohn M, Gonzalez W, Tsurutani B, Lu G (2003) Intense space storms: critical issues and open disputes. J Geophys Res. doi:10.1029/2002JA009722Google Scholar
  8. Daglis IA, Balasis G, Ganushkina N, Metallinou F-A, Palmroth M, Pirjola R, Tsagouri IA (2009) Investigating dynamic coupling in geospace through the combined use of modeling, simulations and data analysis. Acta Geophys. doi:10.2478/s11600-008-0055-5Google Scholar
  9. Ebeling W, Nicolis G (1992) Word frequency and entropy of symbolic sequences: a dynamical Perspective. Chaos Solitons Fractals 2:635–650CrossRefGoogle Scholar
  10. Ebeling W, Steuer R, Titchener M (2001) Partition-based entropies of deterministic and stochastic maps. Stochast Dyn 1:45–61CrossRefGoogle Scholar
  11. Graben P, Kurths J (2003) Detecting subthreshold events in noisy data by symbolic dynamics. Phys Rev Lett 90:100602(1–4).CrossRefGoogle Scholar
  12. Grassberger P, Procaccia I (1983) Estimation of the Kolmogorov entropy from a chaotic signal. Phys Rev A 28:2591–2593CrossRefGoogle Scholar
  13. Hao B-L (1989) Elementary symbolic dynamics and chaos in dissipative systems. World Scientific, SingaporeGoogle Scholar
  14. Henegham C, McDarby G (2000) Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys Rev E 62:6103–6110CrossRefGoogle Scholar
  15. Hurst HE (1951) Long-term storage of reservoirs: an experimental study. Trans Am Soc Civ Eng 116:770–799Google Scholar
  16. Karamanos K (2000) From symbolic dynamics to a digital approach: chaos and transcendence. Lect Notes Phys 550:357–371CrossRefGoogle Scholar
  17. Karamanos K (2001) Entropy analysis of substitutive sequences revisited. J Phys A: Math Gen 34:9231–9241CrossRefGoogle Scholar
  18. Karamanos K, Nicolis G (1999) Symbolic dynamics and entropy analysis of Feigenbaum limit sets. Chaos Solitons Fractals 10(7):1135–1150CrossRefGoogle Scholar
  19. Khinchin AI (1957) Mathematical foundations of information theory. Dover, New York, NYGoogle Scholar
  20. Nicolis G, Gaspard P (1994) Toward a probabilistic approach to complex systems. Chaos Solitons Fractals 4(1):41–57CrossRefGoogle Scholar
  21. Pincus S (1991) Approximate entropy: a complexity measure for biologic time series data. In: Proceedings of IEEE 17th annual northeast bioengineering conference, IEE Press, New York, NY, p 35–36Google Scholar
  22. Pincus S, Keefe D (1992) Quantification of hormone pulsatility via an approximate entropy algorithm. Am J Physiol (Endocrinol Metab) 262: E741–E754Google Scholar
  23. Pincus S, Goldberger A (1994) Physiological time-series analysis: what does regularity quantify? Am J Physiol 266:H1643–H1656Google Scholar
  24. Pincus S, Singer B (1996) Randomness and degree of irregularity. Proc Natl Acad Sci USA 93:2083–2088CrossRefGoogle Scholar
  25. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423Google Scholar
  26. Titchener M, Nicolescu R, Staiger L, Gulliver A, Speidel U (2005) Deterministic complexity and entropy. Fund Inform 64:443–461Google Scholar
  27. Tsallis C (1988) Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys 52:479–487CrossRefGoogle Scholar
  28. Tsallis C (2009) Introduction to nonextensive statistical mechanics, approaching a complex word. Springer, BerlinGoogle Scholar
  29. Vanouplines P (1995) Rescaled range analysis and the fractal dimension of pi. University library, Free University Brussels, Brussels, Belgium.
  30. Wanliss JA (2005) Fractal properties of SYM-H during quiet and active times. J Geophys Res. doi:10.1029/2004JA010544.Google Scholar
  31. Wanliss JA, Dobias P (2007) Space storm as a dynamic phase transition. J Atmos Sol Terr Phys 69:675–684CrossRefGoogle Scholar
  32. Wing S, Johnson J.R. (2010) Introduction to special section on entropy properties and constraints related to space plasma transport. J Geophys Res. doi:10.1029/2009JA014911Google Scholar
  33. Zunino L, Perez D, Kowalski A, Martin M, Garavaglia M, Plastino A, Rosso O (2008) Fractional Brownian motion, fractional Gaussian noise and Tsallis permutation entropy. Physica A 387:6057–6068CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Georgios Balasis
    • 1
  • Ioannis A. Daglis
    • 1
  • Anastasios Anastasiadis
    • 1
  • Konstantinos Eftaxias
    • 2
  1. 1.Institute for Space Applications and Remote Sensing, National Observatory of AthensAthensGreece
  2. 2.Section of Solid State Physics, Department of PhysicsUniversity of AthensAthensGreece

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