Skip to main content
SpringerLink
Log in

The Quantitative “Universal” Label and the Universal Distribution Function for Relative Fluctuations. Qualitative Description of Trendless Random Functions

  • Chapter
  • First Online:
Book cover New Digital Signal Processing Methods

  • 461 Accesses

Abstract

This chapter considers two unexpected applications of the fractional moments. The first application is associated with the distribution of stable points, which are described by defining the quantitative universal label (QUL). The “universal” distribution of stable points that exists in some random sequence can be expressed in the form of the generalised Gaussian distribution (GGD). This GGD is tightly associated with the distribution of the coefficients of some polynomial describing the stable points. If this distribution is general, then a reduced number (six) of stable parameters can describe a broad set of complex phenomena. The QUL is applied for quantitative description of famous transcendental numbers such as π and E (Euler constant). Then examples are provided for economic data and earthquake data. Besides, it is shown how to code valuable information by quantum dots.

Another application of the fractional moments regards the beta-distribution. It is proven that a part of the so-called sequence of the range amplitudes (SRA) after subtraction of the mean value and integration turns out to a bell-like curve that can be fitted by the beta-distribution. It means that the new set of obtained stable parameters can characterise a wide set of fluctuations of trendless sequences. This beta-distribution has remarkable scaling properties and can describe a broad set of long-time series. A nontrivial example is considered in details based on long-time membrane current fluctuations. The application of the fractional moments statistics helps to find the differences between the series containing the hidden output.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H.E. Hurst, Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770 (1951)

    Google Scholar 

  2. R.R. Nigmatullin, G. Smith, Fluctuation-noise spectroscopy and a ‘universal’ fitting function of amplitudes of random sequences. Physica A 320, 291 (2003)

    Article  Google Scholar 

  3. R.R. Nigmatullin, Strongly correlated variables and existence of the universal disctribution function for relative fluctuations. Phys. Wave Phenom. 16(2), 1 (2008)

    Article  Google Scholar 

  4. R.R. Nigmatullin, Universal distribution function for the strongly-correlated fluctuations: general way for description of random sequences. Commun. Nonlinear Sci. Numer. Simul. 15, 637–647 (2010)

    Article  MathSciNet  Google Scholar 

  5. D.J. Hudson, Statistics (Lectures on Elementary Statistics and Probability) (CERN, European Organization for Nuclear Research, Geneva, 1964)

    MATH  Google Scholar 

  6. C. Tsallis, Nonextensive statistics: theoretical, experimental and computational evidences and connections. J. Phys. Braz. 29(1), 1–35 (1999)

    Google Scholar 

  7. V.V. Uchaikin, Automodel anomal diffusion and stable laws. Uspekhi Fizicheskich Nauk 173, 1 (2003). (in Russian)

    Google Scholar 

  8. R.R. Nigmatullin, C. Ionescu, D. Baleanu, NIMRAD: novel technique for respiratory data treatment. J. Signal Image Video Process (2012). https://doi.org/10.1007/s11760-012-0386-1

    Article  Google Scholar 

  9. A. Korn, T. Korn, Mathematical Handbook for Scientists and Engineeres (McGraw–Hill book Company, Inc., New York, Toronto, London, 1961)

    MATH  Google Scholar 

  10. G. Forsythe, M. Malcolm, C.B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Inc, Englewood Cliffs, 1977)

    MATH  Google Scholar 

  11. R.R. Nigmatullin, Recognition of nonextensive statistic distribution by the eigen-coordinates method. Physica A 285, 547 (2000)

    Article  Google Scholar 

  12. M. Al-Hasan, R.R. Nigmatullin, Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method. Renew. Energy 28, 93 (2003)

    Article  Google Scholar 

  13. M.G. Kendall, A. Stuart, The Advanced Theory of Statistics, vol vol. 1 (Ch. Griffin & Co. Ltd., New York, London, Sydney, Toronto, 1962)

    MATH  Google Scholar 

  14. J. Feder, Fractals (Plenum Press, New York and London, 1989)

    MATH  Google Scholar 

  15. R. Yulmetuev, F. Gafarov, P. Hanggi, R.R. Nigmatullin, S. Kayumov, Possibility between earthquakes and explosions seismogram differentiation by discrete stochastic non-Markov processes and local Hurst exponent analysis. Phys. Rev. E. 64, 066132 (2001)

    Article  Google Scholar 

  16. A. Oleinikov, V. Sukhanova, I.R. Nabiev, Fluorescence semiconductive nanocrystalls in biology and medicine. Russ. Nanotechnol. 2(N1–2), 160 (2007). (in Russian)

    Google Scholar 

  17. E. Neher, B. Sakmann, Noise analysis of drug induced voltage clamp currents in denervated frog muscle fibres. J. Physiol. 258, 705–729 (1976)

    Article  Google Scholar 

  18. F.J. Sigworth, The variance of sodium current fluctuations at the node of Ranvier. J. Gen. Physiol. 307, 97–129 (1980)

    Article  Google Scholar 

  19. F.J. Sigworth, Open channel noise. I. Noise in acetylcholine receptor currents suggests conformational fluctuations. Biophys. J. 47, 709–720 (1985)

    Article  Google Scholar 

  20. F.J. Sigworth, Open channel noise. II. A test for coupling between current fluctuations and conformational transitions in the acetylcholine receptor. Biophys. J. 49, 1041–1046 (1986)

    Article  Google Scholar 

  21. P. Läuger, Structural fluctuations and current noise of ionic channels. Biophys. J. 48, 369–373 (1985)

    Article  Google Scholar 

  22. L. Venkataramanan, F.J. Sigworth, Applying hidden Markov models to the analysis of single ion channel activity. Trends Neurosci. 21, 137–145 (1998)

    Article  Google Scholar 

  23. S.F. Traynelisa, F. Jaramilloa, Getting the most out of noise in the central nervous system. Biophys. J. 82, 1930–1942 (2002)

    Article  Google Scholar 

  24. O. Alvarez, C. Gonzalez, R. Latorre, Counting channels: a tutorial guide on ion channel fluctuation analysis. Adv. Physiol. Educ. 26, 327–341 (2002)

    Article  Google Scholar 

  25. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689 (1994)

    Article  Google Scholar 

  26. C.-K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5, 82–87 (1995)

    Article  Google Scholar 

  27. S.M. Ossadnik, S.V. Buldyrev, A.L. Goldberger, S. Havlin, R.N. Mantegna, C.-K. Peng, M. Simons, H.E. Stanley, Correlation approach to identify coding regions in DNA sequences. Biophys. J. 67, 64–70 (1994)

    Article  Google Scholar 

  28. T. Penzel, J.W. Kantelhardt, L. Grote, J.-H. Peter, A. Bunder, Comparison of deterended fluctuation analysis and spectral analysis for heart rate variability in sleep and sleep apnea. I.E.E.E. Trans. Biomed. Eng. 50, 1143–1151 (2003)

    Google Scholar 

  29. M. Jospin, P. Caminal, E.W. Jensen, H. Litvan, M. Vallverdu, R.F. Struys, H.E.M. Vereecke, D.T. Kaplan, Detrended fluctuation analysis of EEG as a measure of depth of anesthesia. I.E.E.E. Trans. Biomed. Eng. 54, 840–846 (2007)

    Google Scholar 

  30. R.L. Burr, C.J. Kirkness, P.H. Mitchell, Detrended fluctuation analysis of the ICP predicts outcome following traumatic brain injury. I.E.E.E. Trans. Biomed. Eng. 55, 2509–2518 (2008)

    Google Scholar 

  31. J.M. Hausdorff, C.-K. Peng, Z. Ladin, J.Y. Wei, A.L. Goldberger, Is walking a random walk? Evidence for long-range correlations in the stride interval of human gait. J. Appl. Physiol. 78, 349–358 (1995)

    Article  Google Scholar 

  32. J.M. Hausdorff, P. Purdon, C.-K. Peng, Z. Ladin, J.Y. Wei, A.L. Goldberger, Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. J. Appl. Physiol. 80, 1448–1457 (1996)

    Article  Google Scholar 

  33. J.W. Kantelhardt, E. Koscielny-Bunde, H.H.A. Rego, S. Havlin, A. Bunde, Detecting long-range correlations with detrended fluctuation analysis. Physica A 295, 441–454 (2001)

    Article  Google Scholar 

  34. C.M. Baleanu, R.R. Nigmatullin, S.S. Cetin, D. Baleanu, S. Ozcelik, New method and treatment technique applied to interband transition in GaAs1-x Px ternary alloys. Centr. Eur. J.Phys. 9, 729–739 (2011)

    Google Scholar 

  35. M.L. Ciurea, S. Lazanu, I. Stavaracher, A.-M. Lepadatu, V. Iancu, M.R. Mitroi, R.R. Nigmatullin, C.M. Baleanu, Stress-induced traps in multilayered structures. J. Appl. Phys. 109, 013717–013716 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Radioelectronics and Informative-Measurement Technics Department, Kazan National Research Technical University named by A.N. Tupolev (KNRTU-KAI), Kazan, Tatarstan Republic, Russia

    Raoul R. Nigmatullin

  2. Department of Electrical and Information Engineering, Polytechnic University of Bari, Bari, Italy

    Paolo Lino & Guido Maione

Authors
  1. Raoul R. Nigmatullin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Lino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guido Maione
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Nigmatullin, R.R., Lino, P., Maione, G. (2020). The Quantitative “Universal” Label and the Universal Distribution Function for Relative Fluctuations. Qualitative Description of Trendless Random Functions. In: New Digital Signal Processing Methods. Springer, Cham. https://doi.org/10.1007/978-3-030-45359-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-45359-6_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-45358-9

  • Online ISBN: 978-3-030-45359-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation