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Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy

Author

Listed:
  • Zunino, L.
  • Pérez, D.G.
  • Kowalski, A.
  • Martín, M.T.
  • Garavaglia, M.
  • Plastino, A.
  • Rosso, O.A.
Abstract
In this work, we analyze two important stochastic processes, the fractional Brownian motion and fractional Gaussian noise, within the framework of the Tsallis permutation entropy. This entropic measure, evaluated after using the Bandt & Pompe method to extract the associated probability distribution, is shown to be a powerful tool to characterize fractal stochastic processes. It allows for a better discrimination of the processes than the Shannon counterpart for appropriate ranges of values of the entropic index. Moreover, we find the optimum value of this entropic index for the stochastic processes under study.

Suggested Citation

  • Zunino, L. & Pérez, D.G. & Kowalski, A. & Martín, M.T. & Garavaglia, M. & Plastino, A. & Rosso, O.A., 2008. "Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6057-6068.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:24:p:6057-6068
    DOI: 10.1016/j.physa.2008.07.004
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    References listed on IDEAS

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    1. Constantino Tsallis & Celia Anteneodo & Lisa Borland & Roberto Osorio, 2003. "Nonextensive statistical mechanics and economics," Papers cond-mat/0301307, arXiv.org.
    2. Keller, K. & Sinn, M., 2005. "Ordinal analysis of time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(1), pages 114-120.
    3. Larrondo, H.A. & González, C.M. & Martín, M.T. & Plastino, A. & Rosso, O.A., 2005. "Intensive statistical complexity measure of pseudorandom number generators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(1), pages 133-138.
    4. Su, Zhi-Yuan & Wu, Tzuyin, 2007. "Music walk, fractal geometry in music," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 418-428.
    5. Capurro, A. & Diambra, L. & Lorenzo, D. & Macadar, O. & Martin, M.T. & Mostaccio, C. & Plastino, A. & Rofman, E. & Torres, M.E. & Velluti, J., 1998. "Tsallis entropy and cortical dynamics: the analysis of EEG signals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 257(1), pages 149-155.
    6. Chstoph Bandt & Faten Shiha, 2007. "Order Patterns in Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(5), pages 646-665, September.
    7. Rosso, O.A & Martin, M.T & Plastino, A, 2003. "Brain electrical activity analysis using wavelet-based informational tools (II): Tsallis non-extensivity and complexity measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 497-511.
    8. Zunino, L. & Pérez, D.G. & Garavaglia, M. & Rosso, O.A., 2007. "Wavelet entropy of stochastic processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 503-512.
    9. McGaughey, Donald R. & Aitken, G.J.M., 2002. "Generating two-dimensional fractional Brownian motion using the fractional Gaussian process (FGp) algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 369-380.
    10. Kalimeri, M. & Papadimitriou, C. & Balasis, G. & Eftaxias, K., 2008. "Dynamical complexity detection in pre-seismic emissions using nonadditive Tsallis entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1161-1172.
    11. Tong, S. & Bezerianos, A. & Paul, J. & Zhu, Y. & Thakor, N., 2002. "Nonextensive entropy measure of EEG following brain injury from cardiac arrest," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(3), pages 619-628.
    12. Tsallis, Constantino & Anteneodo, Celia & Borland, Lisa & Osorio, Roberto, 2003. "Nonextensive statistical mechanics and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 89-100.
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    14. Rosso, Osvaldo A. & De Micco, Luciana & Plastino, A. & Larrondo, Hilda A., 2010. "Info-quantifiers’ map-characterization revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4604-4612.
    15. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2010. "Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1891-1901.
    16. Zhao, Xiaojun & Ji, Mengfan & Zhang, Na & Shang, Pengjian, 2020. "Permutation transition entropy: Measuring the dynamical complexity of financial time series," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
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    19. Samit Paul, 2020. "Time Varying Efficiency in Indian Sectors: An Event Study on Demonetization," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 18(1), pages 103-127, March.

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