Forecasting non-stationary time series by wavelet process modelling

  • Piotr Fryzlewicz
  • Sébastien Van Bellegem
  • Rainer von Sachs
Time Series


Many time series in the applied sciences display a time-varying second order structure. In this article, we address the problem of how to forecast these nonstationary time series by means of non-decimated wavelets. Using the class of Locally Stationary Wavelet processes, we introduce a new predictor based on wavelets and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Finally, we apply the prediction algorithm to a meteorological time series.

Key words and phrases

Local stationarity non-decimated wavelets prediction time-modulated processes Yule-Walker equations 


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  1. Antoniadis, A. and Sapatinas, T. (2003). Wavelet methods for continuous-time prediction using representations of autoregressive processes in Hilbert spaces,J. Multivariate Anal. (to appear).Google Scholar
  2. Berkner, K. and Wells, R. (2002). Smoothness estimates for soft-threshold denoising via translation-invariant wavelet transforms,Appl. Comput. Harmon. Anal.,12, 1–24.MathSciNetCrossRefMATHGoogle Scholar
  3. Brockwell, P. J. and Davis, R. A. (1991).Time Series: Theory and Methods, 2nd ed., Springer, New York.Google Scholar
  4. Calvet, L. and Fisher, A. (2001). Forecasting multifractal volatility,J. Econometrics,105, 27–58.MathSciNetCrossRefMATHGoogle Scholar
  5. Coifman, R. and Donoho, D. (1995). Time-invariant de-noising,Wavelets and Statistics, Vol. 103 (eds. A. Antoniadis and G. Oppenheim), 125–150, Springer, New York.Google Scholar
  6. Dahlhaus, R. (1996a). Asymptotic statistical inference for nonstationary processes with evolutionary spectra,Athens Conference on Applied Probability and Time Series Analysis, Vol. 2, (eds. P. Robinson and M. Rosenblatt), Springer, New York.Google Scholar
  7. Dahlhaus, R. (1996b). On the Kullback-Leibler information divergence of locally stationary processes,Stochastic Process. Appl.,62, 139–168.MathSciNetCrossRefMATHGoogle Scholar
  8. Dahlhaus, R. (1997). Fitting time series models to nonstationary processes,Ann. Statist.,25, 1–37.MathSciNetCrossRefMATHGoogle Scholar
  9. Dahlhaus, R., Neumann, M. H. and von Sachs, R. (1999). Non-linear wavelet estimation of time-varying autoregressive processes,Bernoulli,5, 873–906.MathSciNetCrossRefMATHGoogle Scholar
  10. Daubechies, I. (1992).Ten Lectures on Wavelets, SIAM, Philadelphia.MATHGoogle Scholar
  11. Fryźlewicz, P. (2002). Modelling and forecasting financial log-returns as locally stationary wavelet processes, Research Report, Department of Mathematics, University of Bristol. (http://www.stats. Scholar
  12. Grillenzoni, C. (2000). Time-varying parameters prediction,Ann. Inst. Statist. Math.,52, 108–122.MathSciNetCrossRefMATHGoogle Scholar
  13. Kress, R. (1991).Numerical Analysis, Springer, New York.Google Scholar
  14. Ledolter, J. (1980). Recursive estimation and adaptive forecasting in ARIMA models with time varying coefficients,Applied Time Series Analysis, II (Tulsa, Okla.), 449–471, Academic Press, New York-London.Google Scholar
  15. Mallat, S., Papanicolaou, G. and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes,Ann. Statist.,26, 1–47.MathSciNetCrossRefMATHGoogle Scholar
  16. Mélard, G. and Herteleer-De Schutter, A. (1989). Contributions to the evolutionary spectral theory.J. Time Ser. Anal.,10, 41–63.MathSciNetMATHGoogle Scholar
  17. Nason, G. P. and von Sachs, R. (1999). Wavelets in time series analysis,Philos. Trans. Roy. Soc. London Ser. A,357, 2511–2526.CrossRefMATHGoogle Scholar
  18. Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of evolutionary wavelet spectra,J. Roy. Statist. Soc. Ser. B,62, 271–292.CrossRefGoogle Scholar
  19. Ombao, H., Raz, J., von Sachs, R. and Malow, B. (2001). Automatic statistical analysis of bivariate nonstationary time series,J. Amer. Statist. Assoc.,96, 543–560.MathSciNetCrossRefMATHGoogle Scholar
  20. Ombao, H., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX model of a non-stationary random process,Ann. Inst. Statist. Math.,54, 171–200.MathSciNetCrossRefMATHGoogle Scholar
  21. Philander, S. (1990).El Niño, La Niña and the Southern Oscillation, Academic Press, San Diego.Google Scholar
  22. Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes,J. Roy. Statist. Soc. Ser. B,27, 204–237.MathSciNetMATHGoogle Scholar
  23. Van Bellegem, S. and von Sachs, R. (2003). Forecasting economic time series with unconditional time-varying variance,International Journal of Forecasting (to appear).Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2003

Authors and Affiliations

  • Piotr Fryzlewicz
    • 1
  • Sébastien Van Bellegem
    • 2
  • Rainer von Sachs
    • 2
  1. 1.Department of MathematicsUniversity of BristolBristolUK
  2. 2.Institut de statistiqueUniversité catholique de LouvainBelgium

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