Locally Adaptive Tree-Based Thresholding
Ludger Evers, Timothy Heaton · Journal of Computational and Graphical Statistics · 2009
This article is concerned with the application of thresholding to the estimation of possibly sparse single sequence data observed subject to noise. In such problems, accuracy can be greatly improved by selecting a threshold that adapts to the unknown signal strength. We set out a classification and regression tree approach aimed at partitioning a sequence of inhomogeneous strength into component homogeneous regions where we can independently set a locally adaptive threshold and thus improve estimation. Our method places a mixture prior on each coefficient consisting of an atom of probability at zero and a symmetric probability density. The mixing weight is chosen via Empirical Bayes. The decision on whether a split should occur is based on a score test. Having selected the partitioning and obtained the local mixing weight for each region, estimation is carried out using the posterior median. We evaluate the performance of our method in the single sequence case and for wavelet denoising on both simulated and real data. In the wavelet context we consider two alternative implementations, splitting the coefficients levelwise and splitting the original domain. Our method is cheap to compute and in numerical comparisons our method shows excellent performance when compared with current thresholding techniques. This article has supplementary material online.