Adaptive lifting for nonparametric regression |
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Authors: | Matthew A Nunes Marina I Knight Guy P Nason |
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Institution: | (1) Department of Mathematics, University of Bristol, UK |
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Abstract: | Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2J for some J. These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme
is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data.
A key lifting component is the “predict” step where a prediction of a data point is made. The residual from the prediction
is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing
the kind of prediction according to a criterion. In this way the smoothness of the underlying ‘wavelet’ can be adapted to
the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable
choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods.
We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared
to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting
on the real inductance plethysmography and motorcycle data. |
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Keywords: | Curve estimation Lifting Nonparametric regression Wavelets |
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