Abstract
This paper considers the problem of selecting a robust threshold of wavelet shrinkage. Previous approaches reported in literature to handle the presence of outliers mainly focus on developing a robust procedure for a given threshold; this is related to solving a nontrivial optimization problem. The drawback of this approach is that the selection of a robust threshold, which is crucial for the resulting fit is ignored. This paper points out that the best fit can be achieved by a robust wavelet shrinkage with a robust threshold. We propose data-driven selection methods for a robust threshold. These approaches are based on a coupling of classical wavelet thresholding rules with pseudo data. The concept of pseudo data has influenced the implementation of the proposed methods, and provides a fast and efficient algorithm. Results from a simulation study and a real example demonstrate the promising empirical properties of the proposed approaches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Averkamp, R., Houdré, C.: Wavelets thresholding for non necessarily Gaussian noise: Idealism. Ann. Stat. 31, 110–151 (2003)
Bruce, A., Gao, H.-Y.: Applied Wavelet Analysis with S-PLUS. Springer, New York (1996)
Bruce, A., Donoho, D., Gao, H., Martin, D.: Denoising and robust non-linear wavelet analysis. SPIE Proc. Wavelet Appl. 2242, 325–336 (1994)
Cai, T.T.: Adaptive wavelet estimation: a block thresholding and oracle inequality approach. Ann. Stat. 27, 898–924 (1999)
Cox, D.D.: Asymptotics for M-type smoothing splines. Ann. Stat. 11, 530–551 (1983)
Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothing via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995)
Donoho, D.L., Johnstone, I.M., Hock, J.C., Stern, A.S.: Maximum entropy and the nearly black object. J. R. Stat. Soc. Ser. B 54, 41–81 (1992)
Fan, J., Gijbels, I.: Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. J. R. Stat. Soc. Ser. B 57, 371–394 (1995)
Green, P.J., Silverman, B.W.: Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London (1994)
Johnstone, I.M., Silverman, B.W.: Empirical Bayes selection of wavelet thresholds. Ann. Stat. 33, 1700–1752 (2005)
Kovac, A., Silverman, B.W.: Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Am. Stat. Assoc. 95, 172–183 (2000)
Lee, T.C.M.: Improved smoothing spline regression by combining estimates of different smoothness. Stat. Probab. Lett. 67, 133–140 (2004)
Marron, J.S., Adak, S., Johnstone, I.M., Neumann, M.H., Patil, P.: Exact risk analysis of wavelet regression. J. Comput. Graph. Stat. 7, 278–309 (1998)
Nason, G.P.: Wavelet shrinkage by cross-validation. J. R. Stat. Soc. Ser. B 58, 463–479 (1996)
Nason, G.P., Silverman, B.W.: The discrete wavelet transform in S. J. Comput. Graph. Stat. 3, 163–191 (1994)
Oh, H.-S., Nychka, D.W., Lee, T.C.M.: The role of pseudo data for robust smoothing with application to wavelet regression. Biometrika 94, 893–904 (2007)
Percival, D.B., Walden, A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge (2000)
Sardy, S., Tseng, P., Bruce, A.: Robust wavelet denoising. IEEE Trans. Signal Process. 49, 1146–1152 (2001)
Vidakovic, B.: Statistical Modeling by Wavelets. Wiley, New York (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oh, HS., Kim, D. & Kim, Y. Robust wavelet shrinkage using robust selection of thresholds. Stat Comput 19, 27–34 (2009). https://doi.org/10.1007/s11222-008-9066-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-008-9066-y