Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

NEW SELECTION APPROACH FOR RESOLUTION AND BASIS FUNCTIONS IN WAVELET REGRESSION

  • Received : 2014.04.08
  • Accepted : 2014.05.28
  • Published : 2014.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we propose a new approach to the variable selection problem for a primary resolution and wavelet basis functions in wavelet regression. Most wavelet shrinkage methods focus on thresholding the wavelet coefficients, given a primary resolution which is usually determined by the sample size. However, both a primary resolution and the basis functions are affected by the shape of an unknown function rather than the sample size. Unlike existing methods, our method does not depend on the sample size and also takes into account the shape of the unknown function.

Keywords

References

  1. A. Antoniadi and J. Fan, Regularization of Wavelet Approximations, J. Amer. Statist. Assoc. 96 (2001), 939-955. https://doi.org/10.1198/016214501753208942
  2. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (1992).
  3. D.L. Donoho and I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81 (1994), 425-455. https://doi.org/10.1093/biomet/81.3.425
  4. D.L. Donoho and I.M. Johnstone, Adapting to unknown smoothing via wavelet shrinkage, J. Amer. Statist. Assoc. 90 (1995), 1200-1224. https://doi.org/10.1080/01621459.1995.10476626
  5. I.M. Johnstone and B.W. Silverman, Empirical Bayes selection of wavelet thresholds, Ann. Statist. 33 (2005), 1700-1752. https://doi.org/10.1214/009053605000000345
  6. J.D. Hart, Nonparametric Smoothing and Lack-of-Fit Tests, Berlin: Springer Verlag, (1997).
  7. S.G. Mallat, A theory for multiresolution image denoising schemes using gener-alized Gaussian and complexity priors, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989), 674-693. https://doi.org/10.1109/34.192463
  8. M. Misiti, Y. Misiti, G. Oppenheim and J.M. Poggi, Wavelet toolbox for use with MATLAB, The Math Works Incorporation, (1994).
  9. G.P. Nason, Wavelet shrinkage by cross-validation, J. R. Stat. Soc. Ser. B Stat. Methodol. 58 (1996), 463-479.
  10. C.G. Park, M. Vannucci and J.D. Hart, Bayesian Methods for Wavelet Series in Single-Index Models, J. Comput. Graph. Statist. 14 (4) (2005), 770-794. https://doi.org/10.1198/106186005X79007
  11. C.G. Park, H.S. Oh and H. Lee, Bayesian selection of primary resolution and wavelet basis functions for wavelet regression, Comput. Statist. 23 (2008), 291-302. https://doi.org/10.1007/s00180-007-0055-y
  12. M. Smith and R. Kohn, Nonparametric regression using Bayesian variable selection, J. Econometrics 75 (1996), 317-343. https://doi.org/10.1016/0304-4076(95)01763-1
  13. M. Smith and R. Kohn, A Bayesian approach to nonparametric bivariate regression, J. Amer. Statist. Assoc. 92 (1997), 1522-1535. https://doi.org/10.1080/01621459.1997.10473674
  14. B. Vidakovic and F. Ruggeri, BAMS method: theory and simulations, The Indian Journal of Statistics, Series B, 63 (2001), 234-249.
  15. B. Vidakovic, Statistical Modeling by Wavelets, Wiley, NY, (1999).