Advertisement

Automation and Remote Control

, Volume 80, Issue 1, pp 138–149 | Cite as

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

  • A. V. MakarenkoEmail author
Control Sciences
  • 2 Downloads

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.

Keywords

discrete sequences T-alphabet metric set symbolic analysis financial time series Rossler oscillator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.zbMATHGoogle Scholar
  2. 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.CrossRefzbMATHGoogle Scholar
  3. 3.
    Bowen, R., Metody simvolicheskoi dinamiki (Methods of Symbolic Dynamics), Moscow: Mir, 1979.Google Scholar
  4. 4.
    Gilmore, R. and Lefranc, M., The Topology of Chaos, New York: Wiley-Interscience, 2002.zbMATHGoogle Scholar
  5. 5.
    Hsu, C.S., Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems, New York: Springer-Verlag, 1987.Google Scholar
  6. 6.
    Osipenko, G., Dynamical Systems, Graphs, and Algorithms, Lecture Notes in Mathematics, vol. 1889, Berlin: Springer-Verlag, 2004Google Scholar
  7. 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.CrossRefzbMATHGoogle Scholar
  8. 8.
    Loskutov, A.Yu., Fascination of Chaos, Usp. Fiz. Nauk, 2010, vol. 180, no. 2, pp. 1305–1329.CrossRefGoogle Scholar
  9. 9.
    Makarenko, A.V., The Study of Discrete Mappings in TQ-space. Basic Principles, J. Math. Sci., 2016, no. 2, pp. 19–29.MathSciNetzbMATHGoogle Scholar
  10. 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. 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.CrossRefGoogle Scholar
  12. 12.
    Makarenko, A.V., Estimation of the TQ-complexity of Chaotic Sequences, IFAC-PapersOnLine, 2015, vol. 48, pp. 1049–1055, arXiv:1506.09103.CrossRefGoogle Scholar
  13. 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. 14.
    Levenshtein, V.I., Binary Codes Capable of Correcting Deletions, Insertions, and Reversals, Soviet Phys. Dokl., 1966, vol. 10, pp. 707–710.MathSciNetGoogle Scholar
  15. 15.
    Gusfield, D., Algorithms on Strings, Trees and Sequences, Cambridge: Cambridge Univ. Press, 1997.CrossRefzbMATHGoogle Scholar
  16. 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.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rossler, O.E., Chaos in Abstract Kinetics: Two Prototypes, Bull. Math. Biology, 1977, vol. 39, pp. 275–289.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

Personalised recommendations