Adaptive lifting for nonparametric regression

Abstract

Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2 J 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.

A wavelet lifting approach to long-memory estimation

“…Future work might investigate higher order prediction schemes and larger neighbourhoods; also, the use of adaptive lifting schemes, such as Nunes et al (2006), might provide benefits arising from improved decorrelation. They would also have the advantage of removing the a priori choice of a wavelet basis for our estimator.…”

“…Lifting has been used successfully for nonparametric regression problems and spectral estimation with irregularly sampled observations, see e.g., Trappe and Liu (2000), Nunes et al (2006), Knight and Nason (2009) and . Jansen and Oonincx (2005) give a recent review of lifting.…”

“…Recall that initially, the lifting algorithm works by constructing an initial set of 'integrals' associated to each observation (t i , X i ). One way of doing this is to construct intervals having the endpoints as the midpoints between the initial grid points (t i ) N −1 i=0 , see Nunes et al (2006) and Jansen et al (2009) for more details. We assume that log 2 I r, j r follow a normal distribution with some mean μ and variance σ 2 .…”

A ‘nondecimated’ lifting transform

“…Claypoole et al (1998Claypoole et al ( , 2003 and Piella and Heijmans (2002) use a split-update-predict algorithm through which they adaptively build wavelets. Nunes et al (2006) use the flexibility of the lifting scheme that removes one coefficient at a time in order to build adaptive prediction steps and embed them into the lifting algorithm.…”

“…In our work we shall use the version of the lifting scheme that 'removes one coefficient at a time', as proposed by Jansen et al (2001Jansen et al ( , 2004). This will also allow us to use adaptivity in our nondecimated lifting transform, see Nunes et al (2006) for details on an adaptive lifting scheme.…”