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Wavelet shrinkage of a noisy dynamical system with non-linear noise impact. (English) Zbl 1364.93798

Summary: 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, D. L. Donoho and I. M. Johnstone [Biometrika 81, No. 3, 425–455 (1994; Zbl 0815.62019); Probab. Theory Relat. Fields 99, No. 2, 277–303 (1994; Zbl 0802.62006)] have already defined an optimal filter design in the sense of a minimization of the error made when estimating the pure signal. We set here a different framework where the influence of the noise is non-linear. In particular, we propose a 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 logistic and a Lorenz 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. Finally, besides the tests on an estimated dataset, the method is tested on financial data: oil prices and NOK/USD exchange rate.

MSC:

93E11 Filtering in stochastic control theory
37N35 Dynamical systems in control
60G35 Signal detection and filtering (aspects of stochastic processes)
65T60 Numerical methods for wavelets

Software:

ThreshLab
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References:

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