Advertisement

Wavelet Nonparametric Regression Using Basis Averaging

  • Paul Yau
  • Robert Kohn
Part of the Lecture Notes in Statistics book series (LNS, volume 141)

Abstract

Wavelet methods for nonparametric regression are fast and spatially adaptive. In particular, Bayesian methods are effective in wavelet estimation. Most wavelet methods use a particular basis to estimate the unknown regression function. In this chapter we use a Bayesian approach that averages over several different bases, and also over the Fourier basis, by weighting the estimate from each basis by the posterior probability of the basis. We show that estimators using basis averaging outperform estimators using a single basis and also estimators that first select the basis having the highest posterior probability and then estimate the unknown regression function using that basis.

Keywords

Wavelet Base Basis Average Nonparametric Regression Marginal Likelihood Bayesian Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Clyde, M. Parmigiani, G. and Vidakovic, B. (1997). Multiple shrinkage and subset selection in wavelets. Biometrika, 391-402.Google Scholar
  2. Chipman, H. A., Kolaczyk, E. D. and McCulloch, R. E. (1997) Adaptive Bayesian Wavelet Shrinkage. Journal of the American Statistical Association, 92, 1413–1421.zbMATHCrossRefGoogle Scholar
  3. Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.CrossRefGoogle Scholar
  4. Donoho, D. L. and Johnstone, I. M. (1994) Ideal spatial adaptation via wavelet shrinkage. Biometrika, 81, 425–455.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, 90, 1200–1224.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Donoho, D. L., Johnstone, I. M., Kerkyacherian, G. and Picard, D. (1995) Wavelet shrinkage: asymptopia? (with discussion). Journal of the Royal Statistical Society, Ser. B, 57, 301–370.zbMATHGoogle Scholar
  7. George, E. I. and Foster, D. (1997). Calibration and empirical Bayes variable selection. Preprint.Google Scholar
  8. Kohn, R., Marron, J. S. and Yau, P. (1997) Wavelet estimation using basis selection and basis averaging. Preprint.Google Scholar
  9. Marron, J. S., Adak, S., Johnstone, I. M., Neumann, M. and Patil, P. (1998) Exact risk analysis of wavelet regression. Journal of Computational and Graphical Statistics, 278-309.Google Scholar
  10. Ruppert, D., Sheather, S.J. and Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal fo the American Statistical Association, 90, 1257–1270.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Stone, C.J. (1994) “The use of polynomial splines and their tensor products in multivariate function estimation,” (with discussion), Annals of Statistics, 22, 118–184.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Wahba, G. (1990), Spline models for observational data, Philadelphia: SIAM.zbMATHCrossRefGoogle Scholar
  13. Vidakovic, B. (1998) Nonlinear wavelet shrinkage with Bayes rules and Bayes factors. Journal of the American Statistical Association, 93, 173–179.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Paul Yau
  • Robert Kohn

There are no affiliations available

Personalised recommendations