Skip to main content

Learning Optimal Representations

  • Chapter
  • 185 Accesses

Part of the book series: NATO ASI Series ((NSSB,volume 298))

Abstract

In the analysis of most experimental time series, the question of how to best represent the data has always had an obvious answer - as a time series. However, with the discovery of chaotic attractors, we have learned that the data from many experimental systems can be better represented in a reconstructed state space [1]. This reveals features of the dynamics that were not apparent in the original time series. These state space reconstructions may simply be considered a generalization of the phase space plots used in classical mechanics.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series”, Phys. Rev. Lett., 45, 712–715, 1980.

    Article  Google Scholar 

  2. F. Takens, “Detecting strange attractors in turbulence”, in: D. Rand and L.-S. Young, eds, Dynamical Systems and Turbulence, Warwick, 1980, Lecture notes in mathematics vol. 898, pages 366–381, Berlin, 1981. Springer-Verlag.

    Google Scholar 

  3. T. Sauer, J. A. Yorke, and M. Casdagli, “Embedology”, Preprint, 1991.

    Google Scholar 

  4. A. Fraser and H. Swinney, “Independent coordinates for strange attractors from mutual information”, Phys. Rev. A, 33A, 1134–1140, 1986.

    Article  MathSciNet  Google Scholar 

  5. M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise”, Technical Report LA-UR-91-1010, Los Alamos National Laboratory, 1991.

    Google Scholar 

  6. E. J. Kostelich and J. A. Yorke, “Noise reduction: Finding the simplest dynamical system consistent with the data”, Physica D, 41, 183–196, 1990.

    Article  MathSciNet  Google Scholar 

  7. J. D. Farmer and J. J. Sidorowich, “Optimal shadowing and noise reduction”, Technical Report LA-UR-90-653, Los Alamos National Laboratory, 1990.

    Google Scholar 

  8. J. L. Breeden and N. H. Packard, “Model-based control of nonlinear systems”, preprint, 1991.

    Google Scholar 

  9. J. Holland, Adaptation in Natural and Artificial Systems. University of Michigan Press, 1975.

    Google Scholar 

  10. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, 1989.

    Google Scholar 

  11. N. H. Packard, “A genetic learning algorithm for the analysis of complex data”, Complex Systems, 4, 543, 1990.

    MathSciNet  MATH  Google Scholar 

  12. F. C. Richards, T. P. Meyer, and N. H. Packard, “Extracting cellular automaton rules directly from experimental data”, Physica, 45D, 189–202, 1990.

    Google Scholar 

  13. J. Cremers and A. Hiibler, “Construction of differential equations from experimental data”, Zeit. Naturforsch., 42a, 797–802, 1986.

    Google Scholar 

  14. J. P. Crutchfield and B. S. McNamara, “Equations of motion from a data serie”, J. Complex Sys., 3, 417–452., 1987.

    MathSciNet  Google Scholar 

  15. T. Eisenhammer, A. Hübler, N. Packard, and J. A.S. Kelso, “Modeling experimental time series with ordinary differential equations”, Technical Report CCSR-89-7, Center for Complex Systems Research, 1989.

    Google Scholar 

  16. J. L. Breeden, F. Dinkelacker, and A. Hübler, “Using noise in the modelling and control of dynamical systems”, Phys. Rev. A, 42, 5827, 1990.

    Article  MathSciNet  Google Scholar 

  17. J. L. Breeden and A. Hübler, “Reconstructing equations of motion using unobserved variables”, Phys. Rev. A, 42, 5817–5826, 1990.

    Article  MathSciNet  Google Scholar 

  18. S. M. Omohundro, “Efficient algorithms with neural network behavior”, J. Complex Sys., 1, 273–347, April 1987.

    MathSciNet  MATH  Google Scholar 

  19. J. L. Breeden and N. H. Packard, “Computing the observable number of degrees of freedom from experimental data”, preprint, 1991.

    Google Scholar 

  20. O. Rössler, Phys. Lett., 57A, 397, 1976.

    Google Scholar 

  21. L. Glass and M. C. Mackey, From Clocks to Chaos. Princeton University Press, 1988.

    Google Scholar 

  22. A. M. Fraser, “Information and entropy in strange attractors”, IEEE Trans. Information Theory, 35(2), 245–262, March 1989.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys., 57, 617, 1985.

    Article  MathSciNet  Google Scholar 

  24. T. P. Meyer, F. C. Richards, and N. H. Packard, “Learning algorithm for modeling complex spatial dynamics”, Phys. Rev. Lett., 63, 1735–1738, 1989.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media New York

About this chapter

Cite this chapter

Breeden, J.L., Packard, N.H. (1992). Learning Optimal Representations. In: Bountis, T. (eds) Chaotic Dynamics. NATO ASI Series, vol 298. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3464-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-3464-8_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6534-1

  • Online ISBN: 978-1-4615-3464-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics