Skip to main content
SpringerLink
Log in

Metrization of the T-Alphabet: Measuring the Distance between Multidimensional Real Discrete Sequences

  • Control Sciences
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The measure is proposed of discrete real sequences similarity in the extended space of states. The measure is based on the symbolic CTQ-analysis methods and is applicable to a chaotic and stochastic multidimensional non-equidistant time series as well. The analysis of the proposed metrics is carried out and their basic properties are described. The method efficiency is tested on the model systems differing in complexity and topology of the attractor. The high sensitivity of the developed similarity measures is demonstrated on the example of the financial time series analysis.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Guckenheimer, J. and Holmes, P.J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, New York: Springer, 1990. Translated under the title Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, Moscow: Inst. Komp. Issled., 2002.

    MATH  Google Scholar 

  2. Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995. Translated under the title Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Moscow: Faktorial, 1999.

    Book  MATH  Google Scholar 

  3. Bowen, R., Metody simvolicheskoi dinamiki (Methods of Symbolic Dynamics), Moscow: Mir, 1979.

    Google Scholar 

  4. Gilmore, R. and Lefranc, M., The Topology of Chaos, New York: Wiley-Interscience, 2002.

    MATH  Google Scholar 

  5. Hsu, C.S., Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems, New York: Springer-Verlag, 1987.

    Google Scholar 

  6. Osipenko, G., Dynamical Systems, Graphs, and Algorithms, Lecture Notes in Mathematics, vol. 1889, Berlin: Springer-Verlag, 2004

    Google Scholar 

  7. Anishchenko, V.S., Vadivasova, T.E., Okrokvertskhov, G.A., and Strelkova, G.I., Statistical Properties of Dynamical Chaos, Usp. Fiz. Nauk, 2005, vol. 175, no. 2, pp. 163–179.

    Article  MATH  Google Scholar 

  8. Loskutov, A.Yu., Fascination of Chaos, Usp. Fiz. Nauk, 2010, vol. 180, no. 2, pp. 1305–1329.

    Article  Google Scholar 

  9. Makarenko, A.V., The Study of Discrete Mappings in TQ-space. Basic Principles, J. Math. Sci., 2016, no. 2, pp. 19–29.

    MathSciNet  MATH  Google Scholar 

  10. Makarenko, A.V., Analysis of the Time Structure of Synchronization in Multidimensional Chaotic Systems, Zh. Eksp. Teor. Fiz., 2015, vol. 147, no. 5, pp. 1053–1063.

    Google Scholar 

  11. Makarenko, A.V., TQ-bifurcations in Discrete Dynamical Systems: Analysis of Qualitative Rearrangements of the Oscillation Mode, J. Exper. Theor. Phyz., 2016, vol. 123, no. 4, pp. 666–676.

    Article  Google Scholar 

  12. Makarenko, A.V., Estimation of the TQ-complexity of Chaotic Sequences, IFAC-PapersOnLine, 2015, vol. 48, pp. 1049–1055, arXiv:1506.09103.

    Article  Google Scholar 

  13. Makarenko, A.V., Spacing between Symbols of T-Alphabet and Properties of Discrete Dynamical Systems, Book of Abstracts of Int. Conf. Analysis and Singularities, Moscow: Steklov Mat. Inst., 2012, pp. 78–79.

    Google Scholar 

  14. Levenshtein, V.I., Binary Codes Capable of Correcting Deletions, Insertions, and Reversals, Soviet Phys. Dokl., 1966, vol. 10, pp. 707–710.

    MathSciNet  Google Scholar 

  15. Gusfield, D., Algorithms on Strings, Trees and Sequences, Cambridge: Cambridge Univ. Press, 1997.

    Book  MATH  Google Scholar 

  16. Makarenko, A.V., Multidimensional Dynamic Processes Studied by Symbolic Analysis in Velocity-Curvature Space, Zh. Vychisl. Mat. Mat. Fiz., 2012, vol. 52, no. 7, pp. 1248–1260.

    MathSciNet  MATH  Google Scholar 

  17. Rossler, O.E., Chaos in Abstract Kinetics: Two Prototypes, Bull. Math. Biology, 1977, vol. 39, pp. 275–289.

    Article  MathSciNet  MATH  Google Scholar 

  18. Shilnikov, L.P., Shilnikov, A., Turaev, D., and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. II, in World Scientific Series on Nonlinear Science, Series A, 2001, vol. 5.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    A. V. Makarenko

Authors
  1. A. V. Makarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Makarenko.

Additional information

Russian Text © A.V. Makarenko, 2017, published in Problemy Upravleniya, 2017, No. 2, pp. 2–9.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Makarenko, A.V. Metrization of the T-Alphabet: Measuring the Distance between Multidimensional Real Discrete Sequences. Autom Remote Control 80, 138–149 (2019). https://doi.org/10.1134/S0005117919010120

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919010120

Keywords

Search

Navigation