Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

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

Volumn 62, Issue 2, 2000, Pages 271-292

  Nason, G P ^a   Von Sachs, R ^b   Kroisandt, G ^c  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^c Universität Kaiserslautern ^*  (Germany)

Author keywords

Local stationarity; Non linear wavelet shrinkage; Non stationary time series; Wavelet periodogram; Wavelet processes; Wavelet spectrum

Indexed keywords

EID: 0034354958     PISSN: 13697412     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9868.00231     Document Type: Article

References (36)

1. Abry, P., Goncalves, P. and Flandrin, P. (1995) Wavelets, spectrum analysis and 1/f-processes. Lect. Notes Statist., 103, 15-29.
2. Cambanis, S. and Houdré, C. (1995) On the continuous wavelet transform of second-order random processes. IEEE Trans. Inform. Theory, 41, 628-642.
3. Cambanis, S. and Masry, E. (1994) Wavelet approximation of deterministic and random signals: convergence properties and rates. IEEE Trans. Inform. Theory, 40, 1013-1029.
4. Cheng, B. and Tong, H. (1996) A theory of wavelet representation and decomposition for a general stochastic process. Lect. Notes Statist., 115, 115-129.
5. Chui, C. K. (1997) Wavelets: a Mathematical Tool for Signal Analysis. Philadelphia: Society for Industrial and Applied Mathematics.
6. Coifman, R. R. and Donoho, D. L. (1995) Translation-invariant de-noising. Lect. Notes Statist., 103, 125-150.
7. Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Ann. Statist., 25, 1-37.
8. Daubechies, I. (1992) Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics.
9. Deslauriers, G. and Dubuc, S. (1989) Symmetric iterative interpolation. Constr. Approx., 5, 49-68.
10. Donoho, D. L. (1995) Denoising via soft thresholding. IEEE Trans. Inform. Theory, 41, 613-627.
11. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995) Wavelet shrinkage: asymptopia (with discussion)? J. R. Statist. Soc. B, 57, 301-369.
12. Donoho, D. L., Mallat, S. G. and von Sachs, R. (1998) Estimating covariances of locally stationary processes: rates of convergence of best basis methods. Technical Report 517. Department of Statistics, Stanford University, Stanford.
13. Flandrin, P. (1993) Temps-fréquence. Paris: Hérmes.
14. Gao, H.-Y. (1993) Wavelet estimation of spectral densities in time series analysis. PhD Thesis. University of California, Berkeley.
15. Johnstone, I. M. and Silverman, B. W. (1997) Wavelet threshold estimators for data with correlated noise. J. R. Statist. Soc. B, 59, 319-351.
16. Kawasaki, S. and Shibata, R. (1995) Weak stationarity of a time series with wavelet representation. Jap. J. Indust. Appl. Math., 12, 37-45.
17. Lawton, W. (1993) Applications of complex valued wavelet transforms to subband decomposition. IEEE Trans. Signal Process., 41, 3566-3568.
18. Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattn Anal. Mach. Intell., 11, 674-693.
19. Mallat, S. G., Papanicolaou, G. and Zhang, Z. (1998) Adaptive covariance estimation of locally stationary processes. Ann. Statist., 26, 1-47.
20. Morettin, P. and Chang, Ch. (1995) Wavelet analysis for stationary processes. Technical Report. Departamento de Estatistica, Instituto de Matemática e Estatistica, São Paulo.
21. Nason, G. P. and von Sachs, R. (1999) Wavelets in time series analysis. Phil. Trans. R. Soc. Land. A, 357, 2511-2526.
22. Nason, G. P., Sapatinas, T. and Sawczenko, A. (1999) Wavelet packet modelling of infant sleep state using heart data. To be published.
23. Nason, G. P. and Silverman, B. W. (1994) The discrete wavelet transform in S. J. Comput. Graph. Statist., 3, 163-191.
24. -(1995) The stationary wavelet transform and some statistical applications. Lect. Notes Statist., 103, 281-300.
25. Neumann, M. H. and von Sachs, R. (1995) Wavelet thresholding: beyond the Gaussian i.i.d. Lect. Notes Statist., 103, 301-329.
26. -(1997) Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist., 25, 38-76.
27. Priestley, M. B. (1981) Spectral Analysis and Time Series. London: Academic Press.
28. Reichel, L. and Trefethen, L. N. (1992) Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Lin. Alg. Applic., 162, 153-185.
29. Rioul, O. and Vetterli, M. (1991) Wavelets and signal processing. IEEE Signal Process. Mag., 8, 14-38.
30. von Sachs, R. and MacGibbon, B. (2000) Non-parametric curve estimation by wavelet thresholding with locally stationary errors. Scand. J. Statist., 27, in the press.
31. von Sachs, R., Nason, G. P. and Kroisandt, G. (1997) Adaptive estimation of the evolutionary wavelet spectrum. Technical Report 516. Department of Statistics, Stanford University, Stanford.
32. -(1998) Spectral representation and estimation for locally-stationary wavelet processes. CRM Proc. Lect. Notes, 18, 381-397.
33. von Sachs, R. and Schneider, K. (1996) Wavelet smoothing of evolutionary spectra by nonlinear thresholding. Appl. Comput. Harm. Anal., 3, 268-282.
34. Saito, N. and Beylkin, G. (1993) Multiresolution representations using the autocorrelation functions of compactly supported wavelets. IEEE Trans. Signal Process., 41, 3584-3590.
35. Strang, G. (1993) Wavelet transforms versus Fourier transforms. Bull. (New Ser.) Am. Math. Soc., 28, 288-305.
36. Walter, G. (1994) Wavelets and Other Orthogonal Systems with Applications. Boca Raton: CRC Press.