Multiscale Local Polynomial Models for Estimation and Testing

Abstract

We present a wavelet-like multiscale decomposition based on iterated local polynomial smoothing with scale dependent bandwidths. For reasons of continuity and smoothness, a multiscale smoothing decomposition must be slightly overcomplete, but the redundancy is less than in the nondecimated wavelet transform. Unlike decimated wavelet transforms, multiscale local polynomial decompositions remain numerically stable and the reconstructions are still smooth when the decomposition is applied to time series data on irregular time points. In image denoising, local polynomials outperform nondecimated wavelet transforms, even though the latter have a higher degree of redundancy allowing additional smoothing upon reconstruction. Another benefit from the presented scheme is its ability to construct multiscale decompositions for derivatives of functions. The transform can also be extended towards nonlinear and observation-adaptive data decompositions.