Covariance structure of wavelet coefficients: Theory and models in a Bayesian perspective

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 61, Issue 4, 1999, Pages 971-986

  Vannucci, Marina ^a   Conrradi, Fabio ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b Texas A and M University ^*

Author keywords

Bayesian inference; Nonparametric regression; Wavelet coefficients' correlation structure; Wavelet shrinkage

Indexed keywords

EID: 0033478457     PISSN: 13697412     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9868.00214     Document Type: Article

References (36)

1. Abramovich, F., Sapatinas, T. and Silverman, B. W. (1998) Wavelet thresholding via a Bayesian approach. J. R. Statist. Soc. B, 60, 725-749.
2. Basilevsky, A. (1983) Applied Matrix Algebra in the Statistical Sciences. Amsterdam: North-Holland.
3. Chib, S. and Greenberg, E. (1994) Understanding the Metropolis-Hastings algorithm. Am. Statistn, 49, 327-335.
4. Chipman, H., Kolaczyk, E. and McCulloch, R. (1997) Adaptive Bayesian wavelet shrinkage. J. Am. Statist. Ass., 92, 1413-1421.
5. Clyde, M., Parmigiani, G. and Vidakovic, B. (1998) Multiple shrinkage and subset selection in wavelets. Biometrika, 85, 391-402.
6. Crouse, M., Nowak, R. and Baraniuk, R. (1998) Wavelet-based signal processing using hidden Markov models. IEEE Trans. Signal Process., 46, 886-902.
7. Daubechies, I. (1992) Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics.
8. Dijkerman, R. and Mazumdar, R. (1994) On the correlation structure of the wavelet coefficients of fractional Brownian motion. IEEE Trans. Inform. Theory, 40, 1609-1612.
9. Donoho, D. and Johnstone, I. (1994) Ideal spatial adaption via wavelet shrinkage. Biometrika, 81, 425-455.
10. -(1995) Adapting to unknown smoothness by wavelet shrinkage. J. Am. Statist. Ass., 90, 1200-1224.
11. -(1998) Minimax estimation via wavelet shrinkage. Ann. Statist., 26, 879-921.
12. Donoho, D., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995) Wavelet shrinkage: asymptopia (with discussion)? J. R. Statist. Soc. B, 57, 301-369.
13. Flandrin, P. (1992) Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inform. Theory, 38, 910-917.
14. Geweke, J. (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In Bayesian Statistics 4 (eds J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith), pp. 169-194. Oxford: Oxford University Press.
15. Hall, P. and Patil, P. (1995) Discussion on Wavelet shrinkage: asymptopia? (by D. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard)? J. R. Statist. Soc. B, 57, 355.
16. Hastings, W. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
17. Heidelberger, P. and Welch. P. (1983) Simulation run length control in the presence of an initial transient. Oper. Res., 31, 1109-1144.
18. Johnstone, I. M. and Silverman, B. W. (1997) Wavelet threshold estimators for data with correlated noise. J. R Statist. Soc. B, 59. 319-351.
19. Kovac, A. and Silverman, B. (1998) Extending the scope of wavelet regression methods by coefficient-dependent thresholding. Technical Report 5/1998. University of Dortmund, Dortmund.
20. Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattn Anal. Mach. Intell., 11, 674-693.
21. Math Works (1996) MATLAB Version 5.0.0 .4064 on SOLZ. Matick: Math Works.
22. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953) Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 1087-1092.
23. Meyer, Y. (1992) Wavelets and Operators. Cambridge: Cambridge University Press.
24. Müller. P. (1992) Alternatives to the Gibbs sampling scheme. Technical Report 92-14. Institute of Statistics and Decision Sciences, Duke University, Durham.
25. Nason, G. P. (1996) Wavelet shrinkage using cross-validation. J. R. Statist. Soc. B, 58, 463-479.
26. O'Hagan, A. (1994) Kendall's Advanced Theory of Statistics, vol. 2B. Cambridge: Cambridge University Press.
27. Smith, A. F. M. and Roberts, G. O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Statist. Soc. B, 55, 3-23.
28. Strang, G. (1989) Wavelets and dilation equations: a brief introduction. SIAM Rev., 31, 614-627.
29. Taswell, C. (1995) WavBox 4: a software toolbox for wavelet transforms and adaptive wavelet packet decompositions. In Wavelets and Statistics (eds A. Antomadis and G. Oppenheim). New York: Springer.
30. Tewfik, A. and Kim, M. (1992) Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans. Inform. Theory, 38, 904-909.
31. Tierney, L. (1994) Markov chains for exploring posterior distributions. Ann. Statist., 22, 1701-1728.
32. Vannucci, M. (1996) On the application of wavelets in statistics (in Italian). PhD Thesis. Dipartimento di Statistics G. Parenti, University of Florence, Florence.
33. Vannucci, M., Brown, P. and Fearn, T. (1998) Wavelet analysis of long-memory processes. Technical Report 98-22. University of Kent at Canterbury, Canterbury.
34. Vidakovic. B. (1998) Nonlinear wavelet shrinkage with Bayes rules and Bayes factor. J. Am. Statist. Ass., 93, 173-179.
35. Vidakovic, B. and Müller, P. (1995) Wavelet shrinkage with affine Bayes rules with applications. Technical Report 95-34. Institute of Statistics and Decision Sciences, Duke University, Durham.
36. Wang, Y. (1996) Function estimation via wavelet shrinkage for long memory data. Ann. Statist., 24, 466-484.