A Comparison of Some Thresholding Selection Methods for Wavelet Regression

Authors: Alsaidi M. Altaher, Mohd T. Ismail

Abstract:

In wavelet regression, choosing threshold value is a crucial issue. A too large value cuts too many coefficients resulting in over smoothing. Conversely, a too small threshold value allows many coefficients to be included in reconstruction, giving a wiggly estimate which result in under smoothing. However, the proper choice of threshold can be considered as a careful balance of these principles. This paper gives a very brief introduction to some thresholding selection methods. These methods include: Universal, Sure, Ebays, Two fold cross validation and level dependent cross validation. A simulation study on a variety of sample sizes, test functions, signal-to-noise ratios is conducted to compare their numerical performances using three different noise structures. For Gaussian noise, EBayes outperforms in all cases for all used functions while Two fold cross validation provides the best results in the case of long tail noise. For large values of signal-to-noise ratios, level dependent cross validation works well under correlated noises case. As expected, increasing both sample size and level of signal to noise ratio, increases estimation efficiency.

Keywords: wavelet regression, simulation, Threshold.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1077014

References:

[1] F. Abramovich, F. Sapatinas, and B. W. Silverman ,"Wavelet thresholding via a Bayesian approach," J. R. Stat. Soc., B 60: 725-749, (1998).
[2] F. Abramovich , T. C. Bailey, T. Sapatinas, "Wavelet analysis and its statistical application," The Statistician 49:1-29, 2000).
[3] S. Barber, G. P. Nason, " Real Nonparametric regression using complex wavelets," J. R. Stat. Soc., B 66:927-939, (2004).
[4] M. Clyde, E. I. George, "Empircal Bayes estimation in wavelet nonparametric regression. In Wavelet-Based Models (lecture Notes in Statistcs, Vol. 141)," edit by P. Mller, B. Vidakovice. Oppenheim,309- 322. Springer-Verlage, New York, (1999).
[5] M. Clyde, E. I. George , "Flexible empircal Bayes estimation for wavelets," J. R. Stat. Soc., B 62: 681-698, (2000).
[6] D. L. Donoho, I. M. Johnstone, "Ideal spatial adaptation by wavelet shrinkage," Biometrika 81:425-455, (1994).
[7] D. L. Donoho, I. M. , "Johnstone Adapting to unknown smoothing via wavelet shrinkage," J. Am. Stat. Assoc. 90:1200-1224, (1995).
[8] J. Fan, I. Gijbels, "Data-Driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaption," J. R. Stat. Soc., B 7:371-394, (1995).
[9] P. J. Green, B. W. Silverman, "Nonparametric Regression and Generalized Linear Models, A roughness Penelty Approach," Chapman and Hall, London. (1994).
[10] I. M. Johnstone, B. W. Silverman, "Ebayesthresh: R programs for Empirical Bayes thresholding," J. Stat. Softw. 12: 1-38, (2005a).
[11] I. M. Johnstone, B. W. Silverman, "Empirical Bayes selection of wavelet thresholds," Ann. Stat. 33:1700-1752, (2005b).
[12] D. Kim, H. S. Oh , "CVThresh: R Package for Level Dependent Cross -Validation Thresholding", J. Stat. Softw. (2006).
[13] H. S. Oh, D. Kim, Y. Lee, "Cross-validated wavelet shrinkage. Springer," New York 24:497-512, (2008).
[14] G. P. Nason, "Wavelet shrinkage by cross-validation," J. R. Stat. Soc., B 58:463-479, (1996).
[15] G. P. Nason, B. W. Silverman, "The discrete wavelet transform in S," J. Comput. Graph. Stat. 3:163-191, (1994).
[16] G. P. Nason, "(WaveThresh3 Software. Department of Mathematics, University of Bristol, UK. URL http://www.stats.bris.ac.uk/~wavethresh/.1998).
[17] G. P. Nason ,"Wavelet Methods in Statistics with R," Springer, New York. (2006).
[18] B. W. Silverman , "Density estimation for Statistics and Data Analysis," Chapman and Hall, London, (1986).
[19] B. W. Silverman "Empirical Bayes thresholding: adapting to sparsity when it advantageous to do so," J. Korean Stat. Soc. 36:1-29, (2007).
[20] C. Stein , "Estimation of the mean of a multivariate normal distribution," Ann. Stat. 9:1135-1151, (1981).
[21] M. Stone, "Cross -validatory choice and assessment of statistical predictions," J. R. Stat. Soc., B 36:111-147, (1974).
[22] B. Vidakovic, "Statistical modeling by wavelets," Wiley, New York, (1999a).