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Fractals and Wavelet Fisher’s Information

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Telematics and Computing (WITCOM 2023)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1906))

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Abstract

Fisher’s information measure (FIM) allows to study the complexities associated to random signals and systems and has been used in the literature to study EEG and other physiological signals. In this paper, various time-domain definitions of Fisher’s information are extended to the wavelet domain and closed-form expressions for each definition are obtained for fractal signals of parameter \(\alpha \). Fisher information planes are computed in a range of \(\alpha \) and based on these, characteristics, properties, and the effect of signal length is also identified. Moreover, based on this, a complete characterization of fractals by wavelet Fisher’s information is presented and the potential application of each definition in practical fractal signal analysis is also highlighted.

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References

  1. Goldberger, A.L., Amaral, L.A., Hausdorff, J.M., Ivanov, P.C., Peng, C.K., Stanley, H.E.: Fractal dynamics in physiology: alterations with disease and aging. Proc. Nat. Acad. Sci. 99(suppl 1), 2466–2472 (2002)

    Article  Google Scholar 

  2. Stephen, D.G., Anastas, J.: Fractal fluctuations in gaze speed visual search. Attention Percept. Psychophys. 73(3), 666–677 (2011)

    Article  Google Scholar 

  3. Ducharme, S.W., van Emmerik, R.E.: Fractal dynamics, variability, and coordination in human locomotion. Kinesiol. Rev. 7(1), 26–35 (2018)

    Article  Google Scholar 

  4. Sen, J., McGill, D.: Fractal analysis of heart rate variability as a predictor of mortality: a systematic review and meta-analysis. Chaos Interdisc. J. Nonlin. Sci. 28(7), 072101 (2018)

    Article  Google Scholar 

  5. Frezza, M.: A fractal-based approach for modeling stock price variations. Chaos Interdisc. J. Nonlin. Sci. 28(9), 091102 (2018)

    Article  MathSciNet  Google Scholar 

  6. Bu, L., Shang, P.: Scaling analysis of stock markets. Chaos Interdisc. J. Nonlin. Sci. 24(2), 023107 (2014)

    Article  MathSciNet  Google Scholar 

  7. Gilmore, M., Yu, C.X., Rhodes, T.L., Peebles, W.A.: Investigation of rescaled range analysis, the Hurst exponent, and long-time correlations in plasma turbulence. Phys. Plasmas 9(4), 1312–1317 (2002)

    Article  Google Scholar 

  8. Peng, X.: A discussion on fractal models for transport physics of porous media. Fractals 23(03), 1530001 (2015)

    Article  Google Scholar 

  9. Beran, J., Sherman, R., Taqqu, M.S., Willinger, W.: Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43(2/3/4), 1566–1579 (1995)

    Google Scholar 

  10. Zhang, C., Cui, H., He, Z., Lin, S., Degang, F.: Fractals in carbon nanotube buckypapers. RSC Adv. 6(11), 8639–8643 (2016)

    Article  Google Scholar 

  11. Fernandes, M.A., Rosa, E.A.R., Johann, A.C.B.R., Grégio, A.M.T., Trevilatto, P.C., Azevedo-Alanis, L.R.: Applicability of fractal dimension analysis in dental radiographs for the evaluation of renal osteodystrophy. Fractals 24(01), 1650010 (2016)

    Article  Google Scholar 

  12. Mandelbrot, B.B., Wallis, J.R.: Noah, Joseph, and operational hydrology. Water Resour. Res. 4(5), 909–918 (1968)

    Article  Google Scholar 

  13. Eke, A., et al.: Physiological time series: distinguishing fractal noises from motions. Pflügers Archiv 439(4), 403–415 (2000)

    Article  Google Scholar 

  14. Eke, A., Herman, P., Kocsis, L., Kozak, L.R.: Fractal characterization of complexity in temporal physiological signals. Physiol. Meas. 23(1), R1 (2002)

    Article  Google Scholar 

  15. Delignieres, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., Ninot, G.: Fractal analyses for short time series: a re-assessment of classical methods. J. Math. Psychol. 50(6), 525–544 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49(2), 1685 (1994)

    Article  Google Scholar 

  17. Lin, T.-K., Fajri, H.: Damage detection of structures with detrended fluctuation and detrended cross-correlation analyses. Smart Mater. Struct. 26(3), 035027 (2017)

    Article  Google Scholar 

  18. Kwapień, J., Oświkecimka, P., Drożdż, S.: Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations. Phys. Rev. E 92(5), 052815 (2015)

    Article  Google Scholar 

  19. Ferreira, P.: What detrended fluctuation analysis can tell us about NBA results. Phys. A 500, 92–96 (2018)

    Article  Google Scholar 

  20. Abry, P., Veitch, D.: Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inf. Theory 44(1), 2–15 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stoev, S., Taqqu, M.S., Park, C., Marron, J.S.: On the wavelet spectrum diagnostic for Hurst parameter estimation in the analysis of internet traffic. Comput. Netw. 48(3), 423–445 (2005)

    Article  Google Scholar 

  22. Serinaldi, F.: Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series. Phys. A 389(14), 2770–2781 (2010)

    Article  Google Scholar 

  23. Taqqu, M.S., Teverovsky, V., Willinger, W.: Estimators for long-range dependence: an empirical study. Fractals 3(04), 785–798 (1995)

    Article  MATH  Google Scholar 

  24. Gallant, J.C., Moore, I.D., Hutchinson, M.F., Gessler, P.: Estimating fractal dimension of profiles: a comparison of methods. Math. Geol. 26(4), 455–481 (1994)

    Article  Google Scholar 

  25. Pilgram, B., Kaplan, D.T.: A comparison of estimators for 1f noise. Phys. D Nonlin. Phenom. 114(1–2), 108–122 (1998)

    Article  MATH  Google Scholar 

  26. Stadnitski, T.: Measuring fractality. Front. Physiol. 3, 127 (2012)

    Article  Google Scholar 

  27. Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall, Boca Raton (1994)

    MATH  Google Scholar 

  28. Percival, D.B.: Stochastic models and statistical analysis for clock noise. Metrologia 40, S289–S304 (2003)

    Article  Google Scholar 

  29. Lee, I.W.C., Fapojuwo, A.O.: Stochastic processes for computer network traffic modelling. Comput. Commun. 29, 1–23 (2005)

    Article  Google Scholar 

  30. Veitch, D., Abry, P.: A wavelet based joint estimator of the parameters of long-range dependence. IEEE Trans. Info. Theory 45, 878–897 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Soltani, S., Simard, P., Boichu, D.: Estimation of the self-similarity parameter using the wavelet transform. Signal Process. 84, 117–123 (2004)

    Article  MATH  Google Scholar 

  32. Pesquet-Popescu, B.: Statistical properties of the wavelet decomposition of certain non-gaussian self-similar processes. Signal Process. 75, 303–322 (1999)

    Article  MATH  Google Scholar 

  33. Abry, P., Goncalves, P., Levy-Vehel, J.: Scaling, Fractal and Wavelets. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  34. Flandrin, P.: Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Info. Theory 38, 910–917 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zunino, L., Perez, D.G., Garavaglia, M., Rosso, O.A.: Wavelet entropy of stochastic processes. Phys. A 379, 503–512 (2007)

    Article  Google Scholar 

  36. Perez, D.G., Zunino, L., Martin, M.T., Garavaglia, M., Plastino, A., Rosso, O.A.: Model-free stochastic processes studied with q-wavelet-based information tools. Phys. Lett. A 364, 259–266 (2007)

    Article  MATH  Google Scholar 

  37. Kowalski, A.M., Plastino, A., Casas, M.: Generalized complexity and classical-quantum transition. Entropy 11, 111–123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martin, M.T., Perez, J., Plastino, A.: Fisher information and non-linear dynamics. Phys. A 291, 523–532 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Martin, M.T., Pennini, F., Plastino, A.: Fisher’s information and the analysis of complex signals. Phys. A 256, 173–180 (1999)

    Google Scholar 

  40. Telesca, L., Lapenna, V., Lovallo, M.: Fisher information measure of geoelectrical signals. Phys. A 351, 637–644 (2005)

    Article  MATH  Google Scholar 

  41. Sánchez-Moreno, P., Yánez, R.J., Dehesa, J.S.: Discrete densities and fisher information. In: Proceedings of the 14th International Conference on Difference Equations and Applications. Difference Equations and Applications, pp. 291–298

    Google Scholar 

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Author information

Authors and Affiliations

  1. Universidad Autónoma del Estado de Quintana Roo, Av. Chetumal, Fracc. Prado Norte, 77519, Cancún, Quintana Roo, Mexico

    Julio César Ramírez Pacheco, David Ernesto Troncoso Romero & José Antonio León Borges

  2. Universidad Autónoma del Estado de Quintana Roo, Blvd. Bahía s/n, Del Bosque, 77019, Chetumal, Quintana Roo, Mexico

    Homero Toral Cruz

Authors
  1. Julio César Ramírez Pacheco
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  2. David Ernesto Troncoso Romero
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  3. Homero Toral Cruz
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  4. José Antonio León Borges
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Correspondence to Julio César Ramírez Pacheco .

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Editors and Affiliations

  1. UPIITA - Instituto Politécnico Nacional, México, Mexico

    Miguel Félix Mata-Rivera

  2. ESCOM - Instituto Politécnico Nacional, México, Mexico

    Roberto Zagal-Flores

  3. Universidad Mayor, Santiago, Chile

    Cristian Barria-Huidobro

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Pacheco, J.C.R., Romero, D.E.T., Cruz, H.T., Borges, J.A.L. (2023). Fractals and Wavelet Fisher’s Information. In: Mata-Rivera, M.F., Zagal-Flores, R., Barria-Huidobro, C. (eds) Telematics and Computing. WITCOM 2023. Communications in Computer and Information Science, vol 1906. Springer, Cham. https://doi.org/10.1007/978-3-031-45316-8_6

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  • DOI: https://doi.org/10.1007/978-3-031-45316-8_6

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