JSM 2011 Online Program

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Abstract Details

Activity Number: 510
Type: Contributed
Date/Time: Wednesday, August 3, 2011 : 10:30 AM to 12:20 PM
Sponsor: Section on Nonparametric Statistics
Abstract - #301391
Title: A Lifting Scheme for Multiscale Kernel Smoothing
Author(s): Maarten Jansen*+
Companies: Université Libre de Bruxelles
Address: Boulevard du Triomphe, Brussels, B-1050, Belgium
Keywords: lifting ; kernel ; wavelet ; Laplacian pyramid ; smoothing

The lifting scheme provides a framework for the design of wavelet transforms for observations on nonequidistant covariates. With a design of the transform based on interpolation, the irregularity may invoke numerical problems, when adjacent observations are much closer to each other than to other neighbors. As an alternative, we propose to construct a transform based on smoothing techniques, kernel smoothing being an example. Smoothing is far less sensitive to the errors and the geometry of irregular observations. Unfortunately, smoothing is not an interpolating operation, thereby leading to discontinuous, unsmooth basis functions. The contradiction in the objectives of smoothness and numerical condition can only be solved by the introduction of a slight redundancy in the transform. This leads to a combination of the lifting scheme and Burt-Adelson's Laplacian pyramid. We discuss the design of such a transform, and more specifically the primal moment condition (stating that wavelets should have zero integrals for stability in L2), the dual moment condition (stating that constant inputs should have zero detail coefficients, for the sake of sparsity) and perfect reconstruction.

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