Wavelet Smoothing and Thresholding

Bob Stine

Department of Statistics, Wharton School, Univ of Penn

Background

Some potentially useful references are:

Cody, M. A. (1992). The fast wavelet transform. Dr. Dobbs Journal, 16-27.

The text describes some of the coding techniques needed for implementing the fast wavelet transform. The accompanying program listing show the source, which is also available on-line from Dr. Dobbs.

Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia, PA.

Somewhat more technical that might be appealing the first time, it't full of ideas and the theoretical foundation.

Donoho , D. L. and I. M. Johnstone (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425-455.

Foster, D. P. and R. A. Stine (1997). An information theoretic comparison of model selection criteria. Working paper.

This manuscript explores the relationship between information theory and statistical model selection. Information theory is seen to provide a connection among the various model selection criteria, including AIC, BIC, RIC (hard thresholding), and a recent empirical Bayes version of hard thresholding known as eBIC.

Meyer, Y. (199?). Wavelets and Operators. Cambridge Univ. Press, Cambridge.

Nason, G. P. and B. W. Silverman (1994). The discrete wavelet transform in S. J. Computational and Graphical Statistics, 3 163-191.

The original free S code, latter supplemented by S-Plus's wavelet package. See Bruce et al.

Strang, G. (1989). Wavelets and dilation equations: a brief introduction. SIAM Review, 31, 614-627.

My personal favorite introduction to wavelets. This one makes the connections to matrix operations and filtering quite clear.

----- (1993). Wavelet transforms versus Fourier transforms. Amer. Math. Soc., 28, 288-305.

A useful comparison to traditional Fourier transforms, with examples from image compression. Mentions the FBI "fingerprint contest" for finding the most effective compression of a fingerprint.

----- (1996) textbook.