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Approximating Markov chains.

S M Pincus

    Proceedings of the National Academy of Sciences of the United States of America
    |May 15, 1992
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces finite state Markov chains to approximate complex dynamical systems. These novel chains accurately model system behavior, offering insights into chaos and turbulence.

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    Area of Science:

    • Dynamical Systems Theory
    • Stochastic Processes
    • Information Theory

    Background:

    • Deterministic and stochastic processes often exhibit complex dynamics that are difficult to model.
    • Existing methods may struggle with singular stationary measures or correlating successive time increments.

    Purpose of the Study:

    • To develop a common framework of finite state approximating Markov chains for discrete time deterministic and stochastic processes.
    • To introduce novel approximating chains based on stationary conditional probabilities and Lebesgue measure.
    • To analyze the convergence properties and analytical computability of these chains.

    Main Methods:

    • Development of two types of approximating Markov chains: time-averaged and transient.
    • Analysis of weak convergence of stationary measures for approximating chains to true process measures.

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  • Analytical computation of transient chains and n-time unit probabilities.
  • Explicitly accounting for correlations between successive time increments.

    • Main Results:

    • Stationary measures for both approximating chains converge weakly to true process stationary measures as partition width approaches zero.
    • Transient chains and approximations of all n-time unit probabilities can be computed analytically.
    • Weak continuity is observed with a control parameter for uniformly convergent maps, ensuring moment continuity across bifurcations and chaotic regimes.
    • Approximate entropy is proposed as an information-theoretic rate of entropy for approximating Markov chains.

    Conclusions:

    • The developed finite state Markov chain framework provides a robust method for approximating complex dynamical systems.
    • The analytical computability and continuity properties offer significant advantages for studying bifurcations and chaotic behavior.
    • Approximate entropy is a promising parameter for characterizing turbulence and understanding coarse-graining in physical systems.