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Related Experiment Videos

Symbolic stochastic dynamical systems viewed as binary N-step Markov chains.

O V Usatenko1, V A Yampol'skii, K E Kechedzhy

  • 1A. Ya. Usikov Institute for Radiophysics and Electronics, Ukrainian Academy of Science, 12 Proskura Street, 61085 Kharkov, Ukraine. usatenko@ire.kharkov.ua

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 3, 2004
PubMed
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A new theory models systems with long-range correlations using N-step Markov chains. This research reveals self-similarity in stochastic processes and discusses applications in analyzing written and DNA texts.

Area of Science:

  • Complex Systems
  • Stochastic Processes
  • Information Theory

Background:

  • Systems with long-range correlations are prevalent in nature.
  • Understanding their statistical properties is crucial for various scientific fields.
  • Previous models often lack the flexibility to capture complex dependencies.

Purpose of the Study:

  • To develop a theoretical framework for systems exhibiting long-range correlations.
  • To analyze the statistical properties of such systems, including correlations and distributions.
  • To explore the self-similarity and governing dynamics of these stochastic processes.

Main Methods:

  • Development of a theory based on binary N-step Markov chains.
  • Analytical and numerical computation of correlation functions, distribution functions, and variance.

Related Experiment Videos

  • Investigation of the diffusion Fokker-Planck equation for distribution functions.
  • Analysis of self-similarity and similarity group transformations.

    • Main Results:

    • A model where conditional probabilities depend linearly on preceding symbols.
    • Analytical and numerical derivations of correlation and distribution functions for arbitrary lengths.
    • Identification of self-similarity in the stochastic process.
    • Demonstration that the distribution function approaches a Gaussian form under specific conditions, with non-linear variance dependence on length.
    • Exploration of the diffusion Fokker-Planck equation.

    Conclusions:

    • The developed theory provides a robust framework for analyzing systems with long-range correlations.
    • The findings reveal fundamental properties like self-similarity and Gaussian distribution under certain conditions.
    • The theory has potential applications in analyzing complex data, including written language and DNA sequences.