This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, “the structural intensity”, that computes the “density” of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.
Keywords: Lipschitz singularity, continuous wavelet transform, scale-space representation, zero-crossings, wavelet maxima, feature extraction, non parametric estimation, bagging, landmark-based matching
@article{PS_2005__9__143_0,
author = {Bigot, J\'er\'emie},
title = {A scale-space approach with wavelets to singularity estimation},
journal = {ESAIM: Probability and Statistics},
pages = {143--164},
year = {2005},
publisher = {EDP Sciences},
volume = {9},
doi = {10.1051/ps:2005007},
mrnumber = {2148964},
zbl = {1136.62030},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps:2005007/}
}
TY - JOUR AU - Bigot, Jérémie TI - A scale-space approach with wavelets to singularity estimation JO - ESAIM: Probability and Statistics PY - 2005 SP - 143 EP - 164 VL - 9 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps:2005007/ DO - 10.1051/ps:2005007 LA - en ID - PS_2005__9__143_0 ER -
Bigot, Jérémie. A scale-space approach with wavelets to singularity estimation. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 143-164. doi: 10.1051/ps:2005007
[1] , and, Wavelet estimators in nonparametric regression: a comparative simulation study. J. Statist. Software 6 (2001) 1-83.
[2] and, Detecting abrupt changes by wavelet methods. J. Nonparam. Statist 14 (2001) 7-29. | Zbl
[3] ,, and, Oscillating singularities and fractal functions, in Spline functions and the theory of wavelets (Montreal, PQ, 1996), Amer. Math. Soc., Providence, RI. CRM Proc. Lect. Notes 18 (1999) 315-329.. | Zbl
[4] ,, and, Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl. 4 (1998) 159-174. | Zbl
[5] ,, and, Oscillating singularities on Cantor sets: a grand-canonical multifractal formalism. J. Statist. Phys. 87 (1997) 179-209. | Zbl
[6] , and, The thermodynamics of fractals revisited with wavelets. Physica A 213 (1995) 232-275.
[7] , and, Singularity spectrum of fractal signals: exact results. J. Statist. Phys. 70 (1993) 635-674. | Zbl
[8] , Automatic landmark registration of 1D curves, in Recent advances and trends in nonparametric statistics, M. Akritas and D.N. Politis Eds., Elsevier (2003) 479-496.
[9] , Landmark-based registration of 1D curves and functional analysis of variance with wavelets. Technical Report TR0333, IAP (Interuniversity Attraction Pole network) (2003).
[10] , Bagging Predictors. Machine Learning 24 (1996) 123-140. | Zbl
[11] and, Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 3 (1996) 2384-2398. | Zbl | MR
[12] and, SiZer for exploration of structures in curves. J. Am. Statist. Ass. 94 (1999) 807-823. | Zbl | MR
[13] and space view of curve estimation. Ann. Statist. 28 (2000) 408-428. | Zbl | MR
[14] and, Translation-invariant de-noising, in Wavelets and Statistics, A. Antoniadis and G. Oppenheim, Eds., New York: Springer-Verlag. Lect. Notes Statist. 103 (1995) 125-150. | Zbl | MR
[15] , Ten Lectures on Wavelets. Philadelphia, SIAM (1992). | Zbl | MR
[16] and, Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 (1994) 425-455. | Zbl
[17] and, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Statist. Ass. 90 (1995) 1200-1224. | Zbl | MR
[18] and, Minimax estimation via wavelet shrinkage. Ann. Statist. 26 (1998) 879-921. | Zbl | MR
[19] and, Asymptotic minimality of wavelet estimators with sampled data. Stat. Sinica 9 (1999) 1-32. | Zbl | MR
[20] ,, and, Wavelet shrinkage: Asymptotia? (with discussion). J. R. Statist. Soc. B 57 (1995) 301-337. | Zbl | MR
[21] and, Mode testing via the excess mass estimate. Biometrika 88 (2001) 499-517. | Zbl | MR
[22] and, Searching for Structure in Curve Samples. J. Am. Statist. Ass. 90 (1995) 1179-1188. | Zbl
[23] and, Reconstruction from zero-crossings in scale-space. IEEE Trans. Acoust., Speech, and Signal Proc. 37 (1989) 2111-2130.
[24] , Mathematical Tools for Multifractal Signal Processing. Signal Processing for Multimedia, J.S Byrnes Ed., IOS Press (1999) 111-128. | Zbl
[25] and, Statistical tools to analyze data representing a sample of curves. Ann. Statist. 20 (1992) 1266-1305. | Zbl | MR
[26] , and, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag (1983). | Zbl | MR
[27] , Scale Space Theory in Computer Vision. Kluwer, Boston (1994). | Zbl
[28] , Zero-Crossings of a Wavelet Transform. IEEE Trans. Inform. Theory 37 (1991) 1019-1033. | MR
[29] , A Wavelet Tour of Signal Processing. Academic Press (1998). | Zbl | MR
[30] and, Singularity Detection and Processing with Wavelets. IEEE Trans. Inform. Theory 38 (1992) 617-643. | Zbl | MR
[31] and, Characterization of Signals From Multiscale Egde. IEEE Trans. Pattern Anal. Machine Intelligence 14 (1992) 710-732.
[32] and, Wavelet Transformation Maxima and Multiscale Edges, in Wavelets: A Tutorial in Theory and Applications, C.K. Chui Ed. Boston, Academic Press (1992) 66-104. | Zbl | MR
[33] and, Wavelet Maxima Representation, in Wavelets and Applications, Y. Meyer Ed. New York, Springer-Verlag (1992) 207-284. | Zbl | MR
[34] and, The mode tree: a tool for visualization of nonparametric density features. J. Computat. Graph. Statist. 2 (1993) 51-68.
[35] , and, The bumpy road to the mode forest. J. Comput. Graph. Statist. 7 (1998) 239-251.
[36] ,, and, Décomposition en ondelettes et méthodes comparatives : étude d'une courbe de charge éléctrique. Revue de Statistique Appliquée 17 (1994) 57-77. | Numdam | EuDML
[37] , and, The multifractal formalism revisited with wavelets. Int. J. Bif. Chaos 4 (1994) 245-302. | Zbl | MR
[38] and, Adaptive confidence interval for pointwise curve estimation. Ann. Statist. 28 (2000) 298-335. | Zbl
[39] , Minimax estimation of sharp change points. Ann. Statist. 26 (1998) 1379-1397. | Zbl | MR
[40] and, Curve registration. J. R. Statist. Soc. B 60 (1998) 351-363. | Zbl | MR
[41] and, Functional data analysis. New York, Springer Verlag (1997). | Zbl | MR
[42] and, Bootstrapping with Noise: An Effective Regularization Technique. Connection Science, Special issue on Combining Estimator 8 (1996) 356-372.
[43] , and, On the Asymptotic Convergence of B-Spline Wavelets to Gabor Functions. IEEE Trans. Inform. Theory 38 (1992) 864-872. | Zbl | MR
[44] , Jump and Sharp Cusp Detection by Wavelets. Biometrica 82 (1995) 385-397. | Zbl | MR
[45] and, Alignment of curves by dynamic time warping. Ann. Statist. 25 (1997) 1251-1276. | Zbl | MR
[46] and, Synchronizing sample curves nonparametrically. Ann. Statist. 27 (1999) 439-460. | Zbl | MR
[47] and, Scale-Space Derived From B-Splines. IEEE Trans. on Pattern Analysis and Machine Intelligence 20 (1998) 1040-1055.
[48] , Deformations, Warping and Object Comparison. Tutorial (2000) http://www.cmla.ens-cachan.fr/younes.
[49] and, Scaling Theorems for Zero Crossings. IEEE Trans. Pattern Anal. Machine Intelligence 8 (1986) 15-25. | Zbl
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