Adaptive lifting for nonparametric regression
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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.
KeywordsCurve estimation Lifting Nonparametric regression Wavelets
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- Claypoole R.L., Baraniuk R.G. and Nowak R.D. 1998. Adaptive wavelet transforms via lifting. In Transactions of the International Conference on Acoustics, Speech and Signal Processing. IEEE Trans. Im. Proc. 12: 1513–1516.Google Scholar
- Daubechies I. 1992. Ten Lectures on Wavelets. Philadelphia: SIAM.Google Scholar
- Delouille V., Franke J., and von Sachs R. 2004. Nonparametric stochastic regression with design-adapted wavelets. Sankhya, A, 63: 328–366.Google Scholar
- Delouille V., Simoens J., and von Sachs R. 2004. Smooth design-adapted wavelets for nonparametric stochastic regression. J. Am. Statist. Soc. 99: 643–658.Google Scholar
- Donoho D.L., Johnstone I.M., Kerkyacharian G., and Picard D. 1995b. Wavelet shrinkage: asymptopia? (with discussion) J. Roy. Statist. Soc. B 57: 301–337.Google Scholar
- Friedman J.H. 1984. A variable span scatterplot smoother. Technical Report, No. 5, Laboratory for Computational Statistics, Stanford University, Stanford, CA, USA.Google Scholar
- Green P.J. and Silverman B.W. 1994. Nonparametric regression and generalized linear models. Chapman and Hall: London.Google Scholar
- Jansen M., Nason G.P. and Silverman B.W. 2001. Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients. In Unser, M. and Aldroubi, A. (eds) Wavelet applications in signal and image processing, Proceedings of SPIE 4478: 87–97.Google Scholar
- Jansen M., Nason G.P. and Silverman B.W. 2004. Multivariate nonparametric regression using lifting. Technical Report 04:17, Statistics Group, Department of Mathematics, University of Bristol, UK.Google Scholar
- Johnstone I.M. and Silverman B.W. 2004b. EBayesThresh: R and S-PLUS programs for Empirical Bayes thresholding. Unpublished manuscript (available from the CRAN archive).Google Scholar
- Johnstone I.M. and Silverman B.W. 2005. Empirical Bayes selection of wavelet thresholds. Ann. Statist. 33: (to appear).Google Scholar
- Knight M.I. and Nason G.P. 2004. Improving prediction of hydrophobic segments along a transmembrane protein sequence using adaptive multiscale lifting. Technical Report 04:19, Statistics Group, Department of Mathematics, University of Bristol, UK.Google Scholar
- Kovac A. 1998. Wavelet Thresholding for Unequally Time-Spaced Data. PhD Thesis, University of Bristol.Google Scholar
- Kovac A. and Silverman B.W. 2000. Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Am. Statist. Ass. 95: 172–183.Google Scholar
- Loader C. 1997. Locfit: an introduction. Stat. Comput. Graph. News. 8: 11–17.Google Scholar
- Loader C. 1999. Local regression and likelihood. Springer: New York.Google Scholar
- Nason G.P. and Silverman B.W. 1994. The discrete wavelet transform in S. J. Comp. Graph. Statist. 3: 163–191.Google Scholar
- Nunes M.A. and Nason G.P. 2005. Stopping time in adaptive lifting. Technical Report 05:15, Statistics Group, Department of Mathematics, University of Bristol, UK.Google Scholar
- Percival D.B. and Walden A.T. 2000. Wavelet methods for time series analysis. Cambridge University Press: Cambridge.Google Scholar
- Trappe W. and Liu K.J.R. 2000. Denoising via adaptive lifting schemes. In Proceedings of SPIE, Wavelet applications in signal and image processing VIII, Aldroubi A., Laine M.A. and Unser M.A. (eds), 4119: 302–312.Google Scholar
- Vidakovic B. 1999. Statistical modeling by wavelets. Wiley: New York.Google Scholar