Skip to main content
Log in

A maximum likelihood approach to correlation dimension and entropy estimation

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

To obtain the correlation dimension and entropy from an experimental time series we derive estimators for these quantities together with expressions for their variances using a maximum likelihood approach. The validity of these expressions is supported by Monte Carlo simulations. We illustrate the use of the estimators with a local recording of atrial fibrillation obtained from a conscious dog.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature

  • Abraham, N. B., A. M. Albano, A. Passamante and P. E. Rapp (Eds). 1990.Measures of Complexity and Chaos. New York: Plenum Press.

    Google Scholar 

  • Broggi, G. 1988. Evaluation of dimensions and entropies of chaotic systems.J. opt. Soc. Am. B5, 1020–1028.

    Google Scholar 

  • Denker, M. and G. Keller. 1986. Rigorous statistical procedures for data from dynamical systems.J. stat. Phys. 44, 67–93.

    Article  MATH  MathSciNet  Google Scholar 

  • Eckmann, J.-P. and D. Ruelle. 1985. Ergodic theory of chaos and strange attractors.Rev. mod. Phys. 57, 617–656.

    Article  MATH  MathSciNet  Google Scholar 

  • Ellner, S. 1988. Estimating attractor dimensions from limited data: a new method, with error estimates.Phys. Lett. A133, 128–133.

    Article  Google Scholar 

  • Fraser, A. M. and H. L. Swinney. 1986. Independent coordinates for strange attractors from mutual information.Phys. Rev. A33, 1134–1140.

    MathSciNet  Google Scholar 

  • Grassberger, P. and I. Procaccia. 1983a. Estimation of the Kolmogorov entropy from a chaotic signal.Phys. Rev. A28, 2591–2593.

    Article  Google Scholar 

  • Grassberger, P. and I. Procaccia. 1983b. Measuring the strangeness of strange attractors.Physica 9D, 189–207.

    MathSciNet  Google Scholar 

  • Grassberger, P., R. Badii and A. Politi. 1988. Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors.J. stat. Phys. 51, 135–178.

    Article  MATH  MathSciNet  Google Scholar 

  • Hénon, M. 1976. A two-dimensional mapping with a strange attractor.Commun. math. Phys. 50, 69–77.

    Article  MATH  Google Scholar 

  • Kendall, M. G. and A. Stuart. 1979.The Advanced Theory of Statistics, Vol. 2. London: Griffin.

    MATH  Google Scholar 

  • Kostelich, E. J. and J. A. Yorke. 1990. Noise reduction: finding the simplest dynamical system consistent with the data.Physica D41, 183–196.

    MATH  MathSciNet  Google Scholar 

  • Packard, N. H., J. P. Crutchfield, J. D. Farmer and R. S. Shaw. 1980. Geometry from a time series.Phys. Rev. Lett. 45, 712–716.

    Article  Google Scholar 

  • Ramsey, J. B. and H.-J. Yuan. 1989. Bias and error bars in dimension calculations and their evaluation in some simple models.Phys. Lett. A134, 287–297.

    Article  Google Scholar 

  • Rensma, P. L., M. A. Allessie, W. J. E. P. Lammers, F. I. M. Bonke and M. J. Schalij. 1988. The length of the excitation wave and the susceptibility to reentrant atrial arrhythmias in normal conscious dogs.Circ. Res. 62, 395–410.

    Google Scholar 

  • Ruelle, D. 1990. Deterministic chaos: the science and the fiction.Proc. R. Soc. Lond. A427, 241–248.

    Article  MATH  MathSciNet  Google Scholar 

  • Schuster, H. G. 1988.Deterministic Chaos, an Introduction. Weinheim: VCH.

    Google Scholar 

  • Takens, F. 1981. Detecting strange attractors in turbulence InLecture Notes in Mathematics, Vol. 898, pp. 366–381. Berlin: Springer.

    Google Scholar 

  • Takens, F. 1983. Invariants related to dimension and entropy. InAtas do 13°. Rio de Janeiro: Colóqkio Brasiliero do Matemática.

    Google Scholar 

  • Takens, F. 1985. On the numerical determination of the dimension of an attractor. InLecture Notes in Mathematics, Vol. 1125, pp. 99–106. Berlin: Springer.

    Google Scholar 

  • Theiler, J. 1986. Spurious dimension from correlation algorithms applied to limited time-series data.Phys. Rev. A34, 2427–2432.

    Article  Google Scholar 

  • Theiler, J. 1988. Lacunarity in a best estimator of fractal dimension.Phys. Lett. A133, 195–200.

    Article  MathSciNet  Google Scholar 

  • Theiler, J. 1990a. Estimating fractal dimension.J. opt. Soc. Am. A7, 1055–1073.

    Article  MathSciNet  Google Scholar 

  • Theiler, J. 1990b. Statistical precision of dimension estimators.Phys. Rev. A41, 3038–3051.

    Article  Google Scholar 

  • Wolf, A., J. B. Swift, H. L. Swinney and J. A. Vastano. 1985. Determining Lyapunov exponents from a time series.Physica 16D, 285–317.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Olofsen, E., Degoede, J. & Heijungs, R. A maximum likelihood approach to correlation dimension and entropy estimation. Bltn Mathcal Biology 54, 45–58 (1992). https://doi.org/10.1007/BF02458619

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02458619

Keywords

Navigation