# Continued Fractions in Time Series Forecsting

## Abstract

Through their ergodic properties, continued fractions provide a fascinating example of a dynamical system whose properties we attempt to encode in a library of rules. For an irrational number, randomly selected from a unit interval, the probability distribution of partial quotients can be derived from the ergodic theorem, allowing one to distinguish a random time series from a time series whose elements are drawn from the probability distribution resulting from the ergodic hypothesis. Applications of ergodicity to modular transformations and chaotic cosmology are sketched. In addition to being an object of study, continued fractions are used as a tool to overcome the curse of dimensionality in rule-based forecasting. To this end, we encode the successive (possibly resealed) values of a time series, as the partial quotients of a continued fraction, resulting in a number from the unit interval. The accuracy of a ruled-based system utilizing this coding is investigated to some extent. Qualitative criteria for the applicability of the algorithm are formulated.

## Keywords

Forecast Error Fuzzy Controller Continue Fraction Ergodic Theorem Irrational Number## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Artin, E. (1965): Ein mechanisches System mit quasiergodischen Bahnen, in
*The Collected Papers of Emil Artin*, edited by S. Lang and J. T. Tate ( Addison-Wesley, Reading ), pp. 499–504CrossRefGoogle Scholar - 2.Barrow, J.D. (1981): Chaos in the Einstein equations, Phys. Rev. Lett. 46, 963–966MathSciNetCrossRefGoogle Scholar
- 3.Combs, W.E. and Andrews, J.E. (1998): Combinatorial rule explosion eliminated by a fuzzy rule configuration,
*IEEE Trans*. Fuzzy Systems 6, 111CrossRefGoogle Scholar - 4.Davenport, H. (1983): T
*he Higher Arithmetic: An Introduction to the Theory of Numbers*(Dover, New York ), Ch. I VGoogle Scholar - 5.Jang, J.-S.R. (1993): ANFIS: Adaptive-Network based Fuzzy interference System, IEEE Trasactions on Systems, Man, and Cybernetics,
**25**665–685CrossRefGoogle Scholar - 6.Jan J.K. and Kowng, H.C. (1993): A cryptographic system based upon the continued fractions, Proc. 1993 International Conference on Security Technology (IEEE, Piscataway, NJ ), pp. 219–223.Google Scholar
- 7.Kasabov, N.K. (1996):
*Foundations of Neural Networks, Fuzzy systems, and Knowledge Engineering*( The MIT Press, Cambridge, MA ), Ch. 7MATHGoogle Scholar - 8.Khinchin, A.Ya. (1997):
*Continued Fractions*( Dover, New York )Google Scholar - 9.Lorenz, E.N. (1963): Deterministic nonperiodic flow, J. Atmos. Sc.
**20**, 130–141CrossRefGoogle Scholar - 10.Landau, L.D. and Lifshitz, E.M. (1975) The Classical Theory of Fields (Pergamon Press, New York ), Sec. 118Google Scholar
- 11.Series, C (1981): Non-euclidean geometry, continued fractions, and ergodic theory, Math. Intelligencer
**2**, 24–31MathSciNetGoogle Scholar - 12.Tsoukalas, L.H. and Uhrig, R.E. (1997):
*Fuzzy and Neural Approaches in Engineering*( Wiley, New York ), Sec. 13. 6Google Scholar - 13.Wang, L.X. and Mendel, J.M. (1992): Generating fuzzy numbers by learning from examples. IEEE Trans. Systems, Man and Cybernetics
**22**, 1414–1427MathSciNetCrossRefGoogle Scholar - 14.Wang, L.X. and Mendel, J.M. (1992): Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Trans. Neural Networks,
**3**, 807–813CrossRefGoogle Scholar - 15.Zardecki, A. (1996):
*Rule-based forecasting, in Fuzzy Modelling: Paradigms and Practice*, edited by W. Pedrycz (Kluver, Boston ), pp. 375–391Google Scholar - 16.Zardecki, A. (1995): Fuzzy controllers in nuclear material accounting, Fuzzy Sets and Systems
**74**, 73–79CrossRefGoogle Scholar - 17.Zardecki, A. (1983): Modeling in chaotic relativity, Phys. Rev.
**D28**, 1235–1242MathSciNetGoogle Scholar