Optimal wavelet shrinkage of a noisy dynamical system with non-linear noise impact

Matthieu Garcin () and Dominique Guegan ()
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Matthieu Garcin: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, Natixis Asset Management - SAMS
Dominique Guegan: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique

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Abstract: By filtering wavelet coefficients, it is possible to construct a good estimate of a pure signal from noisy data. Especially, for a simple linear noise influence, Donoho and Johnstone (1994) have already defined an optimal filter design in the sense of a good reconstruction of the pure signal. We set here a different framework where the influence of the noise is non-linear. In particular, we propose an optimal method to filter the wavelet coefficients of a discrete dynamical system disrupted by a weak noise, in order to construct good estimates of the pure signal, including Bayes' estimate, minimax estimate, oracular estimate or thresholding estimate. We present the example of a simple chaotic dynamical system as well as an adaptation of our technique in order to show empirically the robustness of the thresholding method in presence of leptokurtic noise. Moreover, we test both the hard and the soft thresholding and also another kind of smoother thresholding which seems to have almost the same reconstruction power as the hard thresholding.

Keywords: wavelets; dynamical systems; chaos; Gaussian noise; Cauchy noise; thresholding; nonequispaced design; non-linear noise impact (search for similar items in EconPapers)
Date: 2015-10
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01244239
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Published in 2015

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