A Wavelet Transform Approach to Chaotic Short-Term Forecasting

Part of the Intelligent Systems Reference Library book series (ISRL, volume 47)


Chaos theory is widely employed to forecast near-term future values of a time series using data that appear irregular. The chaotic short-term forecasting method is based on Takens’ embedding theorem, which enables us to reconstruct an attractor in a multi-dimensional space using data that appear random but rather are deterministic and geometric in nature. It is difficult to forecast future values of such data based on chaos theory if the information that the data provide cannot be reconstructed through wavelet transformation in a sufficiently low-dimensional space. This paper proposes a method to embed data in a small-dimensional space. This method enables us to abstract the chaotic portion from the focal data and increase forecasting precision.

Chaotic methods are employed to forecast near-term future values of uncertain phenomena. The method makes it possible to restructure an attractor of given timeseries data set in a multidimensional space using Takens’ embedding theory. However, many types of economic time-series data are not sufficiently chaotic. In other words, it is difficult to forecast the future trend of such economic data even based on chaos theory. In this paper, time-series data are divided into wave components using a wavelet transform. Some divided components of time-series data exhibit much more chaotic behavior in the sense of correlation dimension than the original time-series data. The highly chaotic nature of the divided components enables us to precisely forecast the value or the movement of the time-series data in the near future. The up-and-down movement of the TOPICS value is shown to be well predicted by this method, with 70% accuracy.


Chaos theory Short-term forecasting Wavelet transform 


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  1. 1.
    Abhyankar, A., Copeland, L.S., Wong, W.: Nonlinear Dynamics in Real-Time Equity Market Indices: Evidence from the United Kingdom. The Economic Journal 105, 864–880 (1995)CrossRefGoogle Scholar
  2. 2.
    Barnett, W.A., Chen, P.: The Aggregation-Theoretic Monetary Aggregates are Chaotic and Have Strange Attractors: An Econometric Application of Mathematical Chaos. In: Barnett, W., Berndt, E., White, H. (eds.) Dynamic Econometric Modeling. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar
  3. 3.
    Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day, San-Francisco (1976)MATHGoogle Scholar
  4. 4.
    Brock, W.A.: Distinguishing Random and Deterministic Systems: A Bridged Version. Journal of Economic Theory 40, 168–195 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brock, W.A., Sayers, C.L.: Is the Business Cycle Characterized by Deterministic Chaos? Journal of Monetary Economics 22, 71–90 (1988)CrossRefGoogle Scholar
  6. 6.
    Chui, C.K.: Introduction to wavelets. Academic Press, New York (1992)MATHGoogle Scholar
  7. 7.
    Decoster, G., Mitchell, D.: Nonlinear Monetary Dynamics. Journal of Business & Economic Statistics 9(4), 455–461 (1991)Google Scholar
  8. 8.
    Doyne Farmer, J., Sidorowich, J.J.: Exploiting Chaos to Predict the Future and Reduce Noise. In: Lee, Y.C. (ed.) Evolution Learning and Cognition, pp. 277–330. World Scientific, Singapore (1988)Google Scholar
  9. 9.
    Frank, M.Z., Stengo, T.: Measuring the strangeness of gold and silver rates of return. Review of Economic Studies 56, 553–568Google Scholar
  10. 10.
    Grassberger, P., Procaccia, I.: Measuring the Strangeness of Strange Attractors. Physica D 9, 189–208 (1983)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Li, T.Y., Yorke, J.A.: Period three implies chaos. American Mathematical Monthly 82, 985–992 (1975)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lorenz, E.N.: Deterministic non-periodic flow. Journal of the Atmospheric Sciences 20(2), 130–141 (1963)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Matsumoto, Y., Watada, J.: Short-term Prediction by Chaos Method of Embedding Related Data at Same Time. Journal of Japan Industrial Management Association 49(4), 209–217 (1998) (in Japanese)Google Scholar
  14. 14.
    Mees, A.I.: Dynamical systems and tessellations: deteting determinism in data. International Journal of Bifurcation and Chaos 1(4), 777–794 (1991)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Poincare, H.: Sur l’equilbre d’une masse fluide animee’ d’un mouvement de rotation. Acta Mathematica (1885)Google Scholar
  16. 16.
    Ramsey, J.B., Sayers, C.L., Rothman, P.: The Statistical properties of dimension calculations using small data sets: Some economic applications. International Economic Review 31, 991–1020 (1990)CrossRefGoogle Scholar
  17. 17.
    Scheinkman, J.A., LeBaron, B.: Nonlinear Dynamics and Stock Returns. Journal of Business 62(3), 311–337 (1989)CrossRefGoogle Scholar
  18. 18.
    Serizawa, H.: Phenomenon knowledge of chaos. Tokyo Books (1993) (in Japanese)Google Scholar
  19. 19.
    Takens, F.: Detecting Strange Attractors in Turbulence. In: Rand, D.A., Young, L.S. (eds.) Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of EconomicsShimonoseki City UniversityShimonosekiJapan
  2. 2.Graduate School of Information, Production and SystemsWaseda UniversityKitakyushuJapan

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