Statistics and Computing

, Volume 15, Issue 2, pp 83–92 | Cite as

Efficient computation of the discrete autocorrelation wavelet inner product matrix

  • Idris A. Eckley
  • Guy P. Nason


Discrete autocorrelation (a.c.) wavelets have recently been applied in the statistical analysis of locally stationary time series for local spectral modelling and estimation. This article proposes a fast recursive construction of the inner product matrix of discrete a.c. wavelets which is required by the statistical analysis. The recursion connects neighbouring elements on diagonals of the inner product matrix using a two-scale property of the a.c. wavelets. The recursive method is an ↻(log (N)3) operation which compares favourably with the ↻(N log N) operations required by the brute force approach. We conclude by describing an efficient construction of the inner product matrix in the (separable) two-dimensional case.


recursive wavelet relation locally stationary time series autocorrelation wavelets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akay M. (Ed). 1998. Time Frequency and Wavelets in Biomedical Signal Processing, IEEE Press, Piscataway, N.J.Google Scholar
  2. Clements M. and Hendry D. 2001. Forecasting Non-Stationary Economic Time Series, MIT Press, Cambridge, MA.Google Scholar
  3. Dahlhaus R. 1997. Fitting time series models to nonstationary processes, Annals of Statistics 25: 1–37.Google Scholar
  4. Daubechies I. 1988. Orthonormal bases of compactly supported wavelets, Communications in Pure and Applied Mathematics 41: 909–996.Google Scholar
  5. Daubechies I. 1992. Ten Lectures on Wavelets SIAM, Philadelphia.Google Scholar
  6. Deslauriers G. and Dubuc S. 1989. Symmetric iterative interpolation processes, Constructive Approximation 5: 49–68.Google Scholar
  7. Eckley I. 2001. Wavelet Methods for Time Series and Spatial Data, PhD thesis, Department of Mathematics, University of Bristol.Google Scholar
  8. Lio P. and Vannucci M. 2000. Finding pathogenicity islands and gene transfer events in genome data, Bioinformatics 16: 932–940.Google Scholar
  9. Mallat S.G. 1989. A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence 11: 674–693.Google Scholar
  10. Mallat S.G. 1998. A Wavelet Tour of Signal Processing, Academic Press, San DiegoGoogle Scholar
  11. Nason G.P. and Silverman B.W. 1995. The stationary wavelet transform and some statistical applications. In: Antoniadis A. and Oppenheim G. (Eds.), Lecture Notes in Statistics, Vol. 103, Springer-Verlag, pp. 281–300.Google Scholar
  12. Nason G. and von Sachs R. 1999. Wavelets in time series analysis. Phil. Trans. R. Soc. Lond. A. 357:2511–2526.Google Scholar
  13. Nason G., von Sachs R., and Kroisandt G. 2000. Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum, J. Roy. Stat. Soc. Ser. B. 62: 271–292.Google Scholar
  14. Percival D. and Walden A. 2000. Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge.Google Scholar
  15. Priestley M. 1981. Spectral analysis and time series, Academic Press, London.Google Scholar
  16. Saito N. and Beylkin G. 1993. Multiresolution representation using the autocorrelation functions of compactly supported wavelets, IEEE Trans. Sig. Proc. 41: 3584–3590.Google Scholar
  17. Shensa M. 1992. The discrete wavelet transform: wedding the à trous and mallat algorithms, IEEE Trans. Sig. Proc. 40: 2464–2482.Google Scholar
  18. Vidakovic B. 1999. Statistical Modeling by Wavelets, Wiley, New YorkGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Idris A. Eckley
    • 1
  • Guy P. Nason
    • 2
  1. 1.Shell Research Limited and Department of MathematicsUniversity of BristolUK
  2. 2.Department of MathematicsUniversity of BristolBristolUK

Personalised recommendations