Statistics and Computing

, Volume 19, Issue 1, pp 57–71 | Cite as

Adaptive thresholding of sequences with locally variable strength

  • T. J. HeatonEmail author


This paper addresses, via thresholding, the estimation of a possibly sparse signal observed subject to Gaussian noise. Conceptually, the optimal threshold for such problems depends upon the strength of the underlying signal. We propose two new methods that aim to adapt to potential local variation in this signal strength and select a variable threshold accordingly. Our methods are based upon an empirical Bayes approach with a smoothly variable mixing weight chosen via either spline or kernel based marginal maximum likelihood regression. We demonstrate the excellent performance of our methods in both one and two-dimensional estimation when compared to various alternative techniques. In addition, we consider the application to wavelet denoising where reconstruction quality is significantly improved with local adaptivity.


Empirical Bayes Kernel smoothing Locally adaptive Spline smoothing Thresholding Wavelets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramovich, F., Benjamini, Y., Donoho, D.L., Johnstone, I.M.: Adapting to unknown sparsity by controlling the false discovery rate. Ann. Stat. 34, 584–653 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  2. Baraud, Y., Giraud, C., Huet, S.: Gaussian model selection with an unknown variance. Ann. Stat. (2008, to appear) Google Scholar
  3. Beran, R.: Hybrid shrinkage estimators using penalty bases for the ordinal one-way layout. Ann. Stat. 32, 2532–2558 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  4. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont (1984) zbMATHGoogle Scholar
  5. Broyden, C.G.: A new double rank minimization algorithm. Not. Am. Math. Soc. 16, 670 (1969) Google Scholar
  6. Burman, P.: A comparative study of ordinary cross-validation, v-fold cross-validation and the repeated learning-testing methods. Biometrika 76, 503–514 (1989) zbMATHMathSciNetGoogle Scholar
  7. Cai, T.T.: On block thresholding in wavelet regression: Adaptivity, block size, and threshold level. Stat. Sin. 12, 1241–1273 (2002) zbMATHGoogle Scholar
  8. Cai, T.T., Silverman, B.W.: Incorporating information on neighbouring coefficients into wavelet estimation. Sankhyā Ser. B 63, 127–148 (2001) MathSciNetGoogle Scholar
  9. De Boor, C.: A Practical Guide to Splines. Springer, New York (1994) Google Scholar
  10. Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  11. Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  12. Fletcher, R.: A new approach to variable metric algorithms. Int. Bibliogr. Inf. Doc. 13, 317–322 (1970) zbMATHGoogle Scholar
  13. Goldfarb, D.: A family of variable metric methods derived by variational means. Math. Comp. 24, 23–26 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  14. Green, P.J., Silverman, B.W.: Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London (1994) zbMATHGoogle Scholar
  15. Heaton, T.: Wavelets and accompanying applications. Ph.D. thesis, University of Bristol (2005) Google Scholar
  16. Johnstone, I.M., Silverman, B.W.: Needles and straw in haystacks: Empirical Bayes estimates of possible sparse sequences. Ann. Stat. 32, 1594–1649 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  17. Johnstone, I.M., Silverman, B.W.: Ebayesthresh: R programs for empirical Bayes thresholding. J. Stat. Softw. 12, 1–38 (2005a) Google Scholar
  18. Johnstone, I.M., Silverman, B.W.: Empirical Bayes selection of wavelet thresholds. Ann. Stat. 33, 1700–1752 (2005b) zbMATHCrossRefMathSciNetGoogle Scholar
  19. Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of l 2(ℝ). Trans. Am. Math. Soc. 315, 69–87 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  20. Midcourse Space Experiment: (1996–1997) NASA/IPAC infrared science archive.
  21. Nason, G.P.: Wavelet shrinkage using cross-validation. J. R. Stat. Soc. B 58, 463–479 (1996) zbMATHMathSciNetGoogle Scholar
  22. Shanno, D.F.: Conditioning of quasi-newton methods for function minimization. Math. Comp. 24, 647–656 (1970) CrossRefMathSciNetGoogle Scholar
  23. Stone, M.: Cross-validatory choice and assessment of statistical predictions (with discussion). J. R. Stat. Soc. B 36, 114–147 (1974) Google Scholar
  24. Wand, M.P., Jones, M.C.: Kernel Smoothing. Chapman and Hall, London (1995) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of OxfordOxfordUK

Personalised recommendations