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Wavelet Shrinkage

  • Pedro A. Morettin
  • Aluísio Pinheiro
  • Brani Vidakovic
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Wavelet shrinkage provides a simple tool for nonparametric function estimation. It is an active research area where the methodology is based on optimal shrinkage estimators for the location parameters. Some references are Donoho and Johnstone (1994), Donoho et al. (1995), Vidakovic (1999), Antoniadis et al. (2001), Pinheiro and Vidakovic (1997). In this chapter we focus on the simplest, yet most important shrinkage strategy—wavelet thresholding.

Bibliography

  1. F. Abramovich, T. Sapatinas, B.W. Silverman, Wavelet thresholding via a Bayesian approach. J. R. Stat. Soc. Ser. B 60, 725–749 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. A. Antoniadis, J. Bigot, T. Sapatinas, Wavelet estimators in nonparametric regression: a comparative simulation study. J. Stat. Softw. 6, 1–83 (2001)CrossRefGoogle Scholar
  3. A. Bruce, H.-Y. Gao, Understanding WaveShrink: variance and bias estimation. Biometrika 83, 727–745 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. T.T. Cai, L. Brown, Wavelet shrinkage for nonequispaced samples. Ann. Stat. 26, 1783–1799 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. T.T. Cai, L. Brown, Wavelet estimation for samples with random uniform design. Stat. Prob. Lett. 42, 313–321 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. T. Cai, B.W. Silverman, Incorporating information on neighboring wavelet coefficients into wavelet estimators. Sankhya Ser. B 63, 127–148 (2001)zbMATHMathSciNetGoogle Scholar
  7. W. Chang, B. Vidakovic, Wavelet estimation of a baseline signal from repeated noisy measurements by vertical block shrinkage. Comput. Stat. Data Anal. 40, 317–328 (2002)CrossRefzbMATHGoogle Scholar
  8. D. De Canditiis, Wavelet methods for nonparametric regression. PhD thesis, Consiglio Nazionale Delle Ricerche, IAM, Naples, 2001Google Scholar
  9. D. Donoho, Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comput. Harmon. Anal. 1, 100–115 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  10. D. Donoho, I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  11. D. Donoho, I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995a)Google Scholar
  12. D. Donoho, I.M. Johnstone, Adapting to unknown smoothing via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995b)Google Scholar
  13. D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, Wavelet shrinkage: asymptopia? (with discussion). J. R. Stat. Soc. Ser. B 57, 301–369 (1995)zbMATHGoogle Scholar
  14. D. Donoho, I.M. Johnstone, G. Kerkyacharian, D. Pickard, Density estimation by wavelet thresholding. Ann. Stat. 24, 508–539 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. H.-Y. Gao, Choice of thresholds for wavelet shrinkage estimate of the spectrum. J. Time Ser. Anal. 18, 231–251 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. P. Hall, G. Kerkyacharian, D. Picard, On the minimax optimality of blockthresholded wavelet estimators. Stat. Sin. 9, 33–50 (1999)zbMATHGoogle Scholar
  17. W. Härdle, G. Kerkyacharian, D. Picard, A. Tsybakov, Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics, vol. 129 (Springer, New York, 1998)Google Scholar
  18. I.M. Johnstone, B. Silverman, Wavelet threshold estimators for data with correlated noise. J. R. Stat. Soc. Ser. B 59, 319–351 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  19. G. Kerkyachariam, D. Picard, Regression in random design and warped wavelets. Bernoulli 10, 1053–1105 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  20. D. Leporini, J.-C. Pesquet, Wavelet thresholding for a wide class of noise distributions, in EUSIPCO98, Rhodes, September 1998, EUSIPCO (1998), pp. 993–996Google Scholar
  21. G.P. Nason, Wavelet shrinkage using cross-validation. J. R. Stat. Soc. Ser. B 58, 463–479 (1996)zbMATHMathSciNetGoogle Scholar
  22. M.H. Neumann, Spectral density estimation via nonlinear wavelet methods for stationary non-gaussian time series. J. Time Ser. Anal. 17, 601–633 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  23. T. Ogden, E. Parzen, Data dependent wavelet thresholding in nonparametric regression with change points applications. Comput. Stat. Data Anal. 22, 53–70 (1996)CrossRefzbMATHGoogle Scholar
  24. A. O’Hagan, J. Forster, Bayesian inference, in Kendall’s Advanced Theory of Statistics, vol. 2B, 2nd edn. (Wiley, New York, 2004)Google Scholar
  25. J. Opsomer, Y. Wang, Y. Yang, Nonparametric regression with correlated errors. Stat. Sci. 16, 134–153 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  26. D. Pigolli, L.M. Sangalli, Wavelets in functional data analysis: estimation of multidimensional curves and their derivatives. Comput. Stat. Data Anal. 56, 1482–1498 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. A. Pinheiro, B. Vidakovic, Estimating the square root of a density via compactly supported wavelets. Comput. Stat. Data Anal. 25, 399–415 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  28. R.F. Porto, P.A. Morettin, E.C.Q. Aubin, Wavelet regression with correlated errors on a piecewise hölder class. Stat. Prob. Lett. 78, 2739–2743 (2008)CrossRefzbMATHGoogle Scholar
  29. R.F. Porto, P.A. Morettin, E.C.Q. Aubin, Regression with autocorrelated errors using design-adapted haar wavelets. J. Time Ser. Econ. 4, 1–30 (2012)zbMATHMathSciNetGoogle Scholar
  30. N. Saito, Simultaneous noise suppression and signal compression using a library of orthonormal bases and the minimum description length criterion, in Wavelets in Geophysics, ed. by E. Foufoula-Georgiou, P. Kumar (Academic, London, 1994), pp. 299–324CrossRefGoogle Scholar
  31. C. Stein, Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9, 1135–1151 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  32. M. Vannucci, F. Corradi, Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective. J. R. Stat. Soc. Ser. B 61(4), 971–986 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  33. B. Vidakovic, Nonlinear wavelet shrinkage with Bayes rules and Bayes factors. J. Am. Stat. Assoc. 93(441), 173–179 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  34. B. Vidakovic, Statistical Modeling by Wavelets (Wiley, New York, 1999)CrossRefzbMATHGoogle Scholar
  35. B. Vidakovic, P. Müller, Wavelet shrinkage with affine Bayes rules with applications. ISDS Discussion Paper, vol. 95-34 (Duke University, Durham, NC, 1995)Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Pedro A. Morettin
    • 1
  • Aluísio Pinheiro
    • 2
  • Brani Vidakovic
    • 3
  1. 1.Department of StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Department of StatisticsUniversity of CampinasCampinasBrazil
  3. 3.The Wallace H. Coulter Department of Biomedical EngineeringGeorgia Inst Tech & Emory Univ Sch MedAtlantaUSA

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