Optimal wavelet shrinkage of a noisy dynamical system with non-linear noise impact
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
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- Christophe Chorro & Dominique Guégan & Florian Ielpo, 2012.
"Option pricing for GARCH-type models with generalized hyperbolic innovations,"
Taylor & Francis Journals, vol. 12(7), pages 1079-1094, April.
- Christophe Chorro & Dominique Guegan & Florian Ielpo, 2010. "Option pricing for GARCH-type models with generalized hyperbolic innovations," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00469529, HAL.
- Christophe Chorro & Dominique Guegan & Florian Ielpo, 2012. "Option Pricing for GARCH-type Models with Generalized Hyperbolic Innovations," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00511965, HAL.
- Christophe Chorro & Dominique Guegan & Florian Ielpo, 2010. "Option pricing for GARCH-type models with generalized hyperbolic innovations," Documents de travail du Centre d'Economie de la Sorbonne 10023, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
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- Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
- J. Bardet & G. Lang & E. Moulines & P. Soulier, 2000. "Wavelet Estimator of Long-Range Dependent Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 85-99, January.
- D. Bosq & Dominique Guegan, 1995. "Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system," Post-Print halshs-00199345, HAL.
- Antoniadis A. & Fan J., 2001. "Regularization of Wavelet Approximations," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 939-967, September. Full references (including those not matched with items on IDEAS)
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