Calculating Embedding Dimension with Confidence Estimates

  • T. L. CarrollEmail author
  • J. M. Byers
Conference paper
Part of the Understanding Complex Systems book series (UCS)


We describe a method to estimate embedding dimension from a time series. This method includes an estimate of the probability that the dimension estimate is valid. Such validity estimates are not common in algorithms for calculating the properties of dynamical systems. The algorithm described here compares the eigenvalues of covariance matrices created from an embedded signal to the eigenvalues for a covariance matrix of a Gaussian random process with the same dimension and number of points. A statistical test gives the probability that the eigenvalues for the embedded signal did not come from the Gaussian random process.


  1. 1.
    E. Bradley, H. Kantz, Chaos 25, 097610 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    H.D.I. Abarbanel, R. Brown, J.J. Sidorowich, L.S. Tsmring, Rev. Mod. Phy. 65, 1331–1392 (1993)CrossRefGoogle Scholar
  3. 3.
    F. Takens, Detecting Strange Attractors in Turbulence (Springer, New York, 1980)zbMATHGoogle Scholar
  4. 4.
    P. Grassberger, I. Procaccia, Phys. D Nonlinear Phenom. 9, 189–208 (1983)CrossRefGoogle Scholar
  5. 5.
    M.B. Kennel, R. Brown, H.D.I.A. Abarbanel, Phys. Rev. A 45, 3403–3411 (1992)CrossRefGoogle Scholar
  6. 6.
    G.P. King, R. Jones, D.S. Broomhead, Nucl. Phys. B- Proc. Suppl. 2, 379–390 (1987)CrossRefGoogle Scholar
  7. 7.
    A. Maus, J.C. Sprott, Commun. Nonlinear Sci. Numer. Simul. 16, 3294–3302 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Y.I. Molkov, D.N. Mukhin, E.M. Loskutov, A.M. Feigin, G.A. Fidelin, Phys. Rev. E 80, 046207 (2009)CrossRefGoogle Scholar
  9. 9.
    L. Cao, Phys. D Nonlinear Phenom. 110, 43–50 (1997)CrossRefGoogle Scholar
  10. 10.
    T. Buzug, G. Pfister, Phys. Rev. A 45, 7073–7084 (1992)CrossRefGoogle Scholar
  11. 11.
    L.M. Pecora, L. Moniz, J. Nichols, T.L. Carroll, Chaos: Interdiscip. J. Nonlinear Sci. 17, 013110–013119 (2007)Google Scholar
  12. 12.
    J.F. Restrepo, G. Schlotthauer, Phys. Rev. E 94, 012212 (2016)CrossRefGoogle Scholar
  13. 13.
    S. Kullback, R.A. Leibler, Ann. Math. Stat. 22, 79–86 (1951)CrossRefGoogle Scholar
  14. 14.
    T.L. Carroll, J.M. Byers, Phys. Rev. E 93, 042206 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Theiler, Phys. Rev. A 34, 2427–2432 (1986)CrossRefGoogle Scholar
  16. 16.
    J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Phys. D 58, 77–94 (1992)CrossRefGoogle Scholar
  17. 17.
    J. Wishart, Biometrika 20A, 32–52 (1928)CrossRefGoogle Scholar
  18. 18.
    V.A. Marchenko, L.A. Pastur, Math. USSR-Sb. 1, 457–483 (1967)CrossRefGoogle Scholar
  19. 19.
    A. Provenzale, A.R. Osborne, in Dynamics and Stochastic Processes: Theory and Applications, ed. by R. Lima, L. Streit, R.V. Mendes, vol 355 (1990), pp. 260–275Google Scholar
  20. 20.
    O.E. Rossler, Z. Naturforsch 38a, 788–801 (1983)Google Scholar
  21. 21.
    T.L. Carroll, J.M. Byers, Chaos: Interdiscip. J. Nonlinear Sci. 27, 023101 (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.US Naval Research Lab, Code 6392Washington, DCUSA
  2. 2.US Naval Research Lab, Code 6395Washington, DCUSA

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