Multiscale Poisson Data Smoothing

A framework for non‐linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves is introduced, which combines the advantages of the Haar–Fisz transform with wavelet smoothing and (Bayesian) multiscales likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families.

Abstract

Summary The paper introduces a framework for non-linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves. The key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient. Within this framework, most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used. Any family of wavelet transforms (orthogonal, biorthogonal or second generation) can be incorporated in this framework. Our second contribution is to propose a Bayesian shrinkage approach with an original prior for coefficients of this decomposition. As such, the method combines the advantages of the Haar–Fisz transform with wavelet smoothing and (Bayesian) multiscale likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families. Simulations show an important reduction in average squared error of the output, compared with the present techniques of Anscombe or Fisz variance stabilization or multiscale likelihood modelling.