Thresholding of Wavelet Coefficients as Multiple Hypotheses Testing Procedure

Abstract

Given noisy signal, its finite discrete wavelet transform is an estimator of signal’s wavelet expansion coefficients. An appropriate thresholding of coefficients for further reconstruction of de-noised signal plays a key-role in the wavelet decomposition/reconstruction procedure. [DJ1] proposed a global threshold $\lambda = \sigma \sqrt{2\log{n}}$ and showed that such a threshold asymptotically reduces the expected risk of the corresponding wavelet estimator close to the possible minimum. To apply their threshold for finite samples they suggested to always keep coefficients of the first coarse $j_0$ levels. We demonstrate that the choice of $j_0$ may strongly affect the corresponding estimators. Then, we consider the thresholding of wavelet coefficients as a multiple hypotheses testing problem and use the False Discovery Rate (FDR) approach to multiple testing of [BH1]. The suggested procedure controls the expected proportion of incorrectly kept coefficients among those chosen for the wavelet reconstruction. The resulting procedure is inherently adaptive, and responds to the complexity of the estimated function. Finally, comparing the proposed FDR-threshold with that fixed global of Donoho and Johnstone by evaluating the relative Mean-Square-Error across the various test-functions and noise levels, we find the FDR-estimator to enjoy robustness of MSE-efficiency.