Abstract
Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new ‘nondecimated’ lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a ‘best’ representation, which we shall explore.
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Abramovich, F., Bailey, T., Sapatinas, T.: Wavelet analysis and its statistical applications. J. Roy. Stat. Soc. D 49, 1–29 (2000)
Ballester, P.J., Carter, J.N.: Real-parameter genetic algorithms for finding multiple optimal solutions in multi-modal optimization. In: Lecture Notes in Computer Science, vol. 2723, pp. 706–717. Springer, Berlin (2003)
Barber, S., Nason, G.P.: Real nonparametric regression using complex wavelets. J. R. Stat. Soc. B 66, 927–939 (2004)
Chiann, C., Morettin, P.A.: A wavelet analysis for time series. J. Nonparametr. Stat. 10, 1–46 (1999)
Claypoole, R.L., Baraniuk, R.G., Nowak, R.D.: Adaptive wavelet transforms via lifting. In: Transactions of the International Conference on Acoustics, Speech and Signal Processing. IEEE Trans. Image Process. 12, 1513–1516 (1998)
Claypoole, R.L., Davis, G.M., Sweldens, W., Baraniuk, R.G.: Nonlinear wavelet transforms for image coding via lifting. IEEE Trans. Image Process. 12, 1449–1459 (2003)
Coifman, R.R., Donoho, D.L.: Translation-invariant de-noising. In: Antoniadis, A., Oppenheim, G. (eds.) Wavelets and Statistics. Lecture Notes in Statistics, vol. 103, pp. 125–150. Springer, Berlin (1995)
Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
Deb, K., Agrawal, S.: Understanding interactions among genetic algorithm parameters. In: Banzhaf, W., Reeves, C.R. (eds.) Foundations of Genetic Algorithms, vol. 5, pp. 265–286 (1999)
Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Soc. 90, 1200–1224 (1995)
Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, P.: A real-time algorithm for signal analysis with the help of the wavelet transform. In: Combers, J.M., Grossmann, A., Tchamitchian, P. (eds.) Wavelets: Time-Frequency Methods and Phase Space, pp. 286–297 (1989)
Jansen, M., Nason, G.P., Silverman, B.W.: Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients. In: Unser, M., Aldroubi, A. (eds.) Wavelet applications in signal and image processing. Proceedings of SPIE, vol. 4478, pp. 87–97 (2001)
Jansen, M., Nason, G.P., Silverman, B.W.: Multivariate nonparametric regression using lifting. Technical Report 04:17, Statistics Group, Department of Mathematics, University of Bristol, UK (2004)
Johnstone, I.M., Silverman, B.W.: Needles and hay in haystacks: empirical Bayes estimates of possibly sparse sequences. Ann. Stat. 32, 1594–1649 (2004a)
Johnstone, I.M., Silverman, B.W.: EbayesThresh: R programs for empirical Bayes thresholding. J. Stat. Softw. 12, 1–38 (2004b)
Johnstone, I.M., Silverman, B.W.: Empirical Bayes selection of wavelet thresholds. Ann. Stat. 33, 1700–1752 (2005)
Lee, C.S., Lee, C.K., Yoo, K.Y.: New lifting based structure for undecimated wavelet transform. Electron. Lett. 36, 1894–1895 (2000)
Lucasius, C.B., Kateman, G.: Understanding and using genetic algorithms, part 1: concepts, properties and context. Chemom. Intell. Lab. Syst. 19, 1–33 (1993)
Lucasius, C.B., Kateman, G.: Understanding and using genetic algorithms, part 2: representation, configuration and hybridization. Chemom. Intell. Lab. Syst. 25, 99–145 (1994)
Nason, G.P., Silverman, B.W.: The discrete wavelet transform in S. J. Comput. Graph. Stat. 3, 163–191 (1994)
Nason, G.P., Silverman, B.W.: The Stationary wavelet transform and some statistical applications. In: Antoniadis, A., Oppenheim, G. (eds.) Wavelets and Statistics. Lecture Notes in Statistics, vol. 103, pp. 281–300. Springer, Berlin (1995)
Nason, G.P., von Sachs, R., Kroisandt, G.: Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. B 62, 271–292 (2000)
Nunes, M.A., Knight, M.I., Nason, G.P.: Adaptive lifting in nonparametric regression. Stat. Comput. 16, 143–159 (2006)
Nunes, M.A., Nason, G.P.: Stopping time in adaptive lifting. Technical Report 05:15, Statistics Group, Department of Mathematics, University of Bristol, UK (2005)
Percival, D.B.: On estimation of the wavelet variance. Biometrika 82, 619–631 (1996)
Percival, D.B., Walden, A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge (2000)
Pesquet, J.C., Krim, H., Carfantan, H.: Time invariant orhtonormal wavelet representations. IEEE Trans. Signal Process. 44, 1964–1970 (1996)
Piella, G., Heijmans, H.J.A.M.: Adaptive lifting schemes with perfect reconstruction. IEEE Trans. Signal Process. 50, 1620–1630 (2002)
Sweldens, W.: Wavelets and the lifting scheme: a 5 minute tour. Z. Angew. Math. Mech. 76, 41–44 (1996)
Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29, 511–546 (1998)
Trappe, W., Liu, K.J.R.: Denoising via adaptive lifting schemes. In: Aldroubi, A., Laine, M.A., Unser, M.A. (eds.) Wavelet Applications in Signal and Image Processing VIII. Proceedings of SPIE, vol. 4119, pp. 302–312 (2000)
Vidakovic, B.: Statistical Modeling by Wavelets. Wiley, New York (1999)
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Knight, M.I., Nason, G.P. A ‘nondecimated’ lifting transform. Stat Comput 19, 1–16 (2009). https://doi.org/10.1007/s11222-008-9062-2
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DOI: https://doi.org/10.1007/s11222-008-9062-2