Wavelet Shrinkage via Peaks over Threshold

Wavelet shrinkage is a non parametric technique used in curve estimation. The idea is to shrink wavelet coefficients towards zero using statistical methods. More specifically, a threshold value is chosen and wavelet coefficients whose absolute values exceed that threshold are kept while others are removed. This has the effect of both reducing the noise contribution and compressing the original data while keeping a good quality of approximation. A key step in the wavelet shrinkage procedure is the choice of the threshold. In this paper, we present a data driven method for choosing the threshold based on the Peaks Over Threshold modelling of extreme values. Particular features of our method include estimating the extreme value index of the noise via a Generalised Pareto Distribution fit to noisy wavelet transforms. This proposal differs from previous thresholding method based on extreme value theory for Gaussian processes. Results show that the GPD approach outperforms traditional thresholding methods (benchmarks) in cases of heavy-tailed noise perturbations while being very competitive for Gaussian noise.