Statistics and Computing

, Volume 9, Issue 1, pp 65–75 | Cite as

Wavelet shrinkage for unequally spaced data

  • Sylvain Sardy
  • Donald B. Percival
  • Andrew G. Bruce
  • Hong-Ye Gao
  • Werner Stuetzle


Wavelet shrinkage (WaveShrink) is a relatively new technique for nonparametric function estimation that has been shown to have asymptotic near-optimality properties over a wide class of functions. As originally formulated by Donoho and Johnstone, WaveShrink assumes equally spaced data. Because so many statistical applications (e.g., scatterplot smoothing) naturally involve unequally spaced data, we investigate in this paper how WaveShrink can be adapted to handle such data. Focusing on the Haar wavelet, we propose four approaches that extend the Haar wavelet transform to the unequally spaced case. Each approach is formulated in terms of continuous wavelet basis functions applied to a piecewise constant interpolation of the observed data, and each approach leads to wavelet coefficients that can be computed via a matrix transform of the original data. For each approach, we propose a practical way of adapting WaveShrink. We compare the four approaches in a Monte Carlo study and find them to be quite comparable in performance. The computationally simplest approach (isometric wavelets) has an appealing justification in terms of a weighted mean square error criterion and readily generalizes to wavelets of higher order than the Haar.

Nonparametric Regression Wavelet Transform WaveShrink 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Antoniadis, A., Gregoire, G. and McKeague, I. W. (1994) Wavelet Methods for Curve Estimation. Journal of the American Statistical Association, 89(428), 1340–53.Google Scholar
  2. Antoniadis, A., Gregoire, G. and Vial, P. (1997) Random Design Wavelet Curve Smoothing. Statistics & Probability Letters, 35, 225–32.Google Scholar
  3. Bruce, A. G. and Gao, H.-Y. (1996) Understanding WaveShrink: Variance and Bias Estimation. Biometrika, 83(4), 727–45.Google Scholar
  4. Delyon, B. and Juditsky, A. (1995) Estimating Wavelet Coeffi-cients. In Antoniadis, A. and Oppenheim, G., editors, Wavelets and Statistics, pages 151–68. Springer-Verlag, New York.Google Scholar
  5. Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1995) Wavelet Shrinkage: Asymptopia? Journal of the Royal Statistical Society, Series B, 57, 301–69. (with discussion).Google Scholar
  6. Donoho, D. L. and Johnstone, I. M. (1994) Ideal Spatial Adap-tation via Wavelet Shrinkage. Biometrika, 81, 425–55.Google Scholar
  7. Donoho, D. L. and Johnstone, I. M. (1995) Adapting to Un-known Smoothness via Wavelet Shrinkage. Journal of the American Statistical Association, 90(432), 1200–24.Google Scholar
  8. Foster, G. (1996) Wavelets for Period Analysis of Unequally Sampled Time Series. Astronomical Journal, 112(4), 1709–29Google Scholar
  9. Hall, P. and Turlach, B. A. (1997) Interpolation Methods for Nonlinear Wavelet Regression with Irregularly Spaced De-sign. Annals of Statistics, 25(5), 1912–25.Google Scholar
  10. Johnstone, I. M. and Silverman, B. W. (1997) Wavelet Threshold Estimators for Data with Correlated Noise. Journal of the Royal Statistical Society, Series B, 59(2), 319–51.Google Scholar
  11. Kovac, A. and Silverman, B. W. (1998) Extending the Scope of Wavelet Regression Methods by Coefficient-Dependent Thresholding. Technical Report SFB 474, University of Dortmund.Google Scholar
  12. Mallat, S. (1989) A Theory for Multiresolution Signal Decom-position: the Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 674–93.Google Scholar
  13. Okabe, A., Boots, B. and Sugihara, K. (1992) Spatial Tessela-tions: Concepts and Applications of Voronoi Diagrams. John Wiley, New York.Google Scholar
  14. Scargle, J. D. (1997a) Astronomical Time Series Analysis: New Methods for Studying Periodic and Aperiodic Systems. In Maoz, D., Sternberg, A. and Leibowitz, E. M., editors, Astronomical Time Series. Kluwer Academic Publishers, Dordrecht.Google Scholar
  15. Scargle, J. D. (1997b) Wavelet Methods in Astronomical Time Series Analysis. In Rao, T. S., Priestley, M. B. and Lessi, O., editors, Applications of Time Series Analysis in Astronomy and Meteorology, pages 226–48. Chapman and Hall, London.Google Scholar
  16. Sweldens, W. (1995) The Lifting Scheme: a New Philosophy in Biorthogonal Wavelet Constructions. In SPIE proceedings, Signal and Image Processing III, volume 2569, pages 68–79, San Diego, CA.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Sylvain Sardy
  • Donald B. Percival
  • Andrew G. Bruce
  • Hong-Ye Gao
  • Werner Stuetzle

There are no affiliations available

Personalised recommendations