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Theses

On concentration, noise and entropy estimation in dynamical systems

Abstract : This thesis is divided into three parts. In the first part we briefly describe the class of dynamical systems considered. We also give some known results on the study of fluctuations of observables in dynamical systems such as the central limit theorem, large deviations and concentration inequalities. In the second part we study dynamical systems perturbed by observational noise. We prove that if a dynamical system satisfies a concentration inequality then the system with observational noise also satisfies a concentration inequality. We apply these inequalities to obtain fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension. Next, we study the work of S. Lalley on the problem of signal recovery. Given a time series of a chaotic dynamical system with observational noise, one can effectively eliminate the noise in average by using Lalley's algorithm. A chapter of this thesis is devoted to the proof of consistency of that algorithm. We end up the second part with a numerical quest for the best parameters of Lalley's algorithm. The third part is devoted to entropy estimation in one-dimensional Gibbs measures. We study the fluctuations of two entropy estimators. The first one is based on the empirical frequencies of observation of typical blocks. The second is based on the time a typical orbit takes to hit an independent typical block. We apply concentration inequalities to obtain bounds on the fluctuation of these estimators.
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Contributor : Cesar Maldonado <>
Submitted on : Tuesday, September 25, 2012 - 6:27:02 PM
Last modification on : Wednesday, March 27, 2019 - 4:02:05 PM
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Cesar Maldonado. On concentration, noise and entropy estimation in dynamical systems. Mathematical Physics [math-ph]. Ecole Polytechnique X, 2012. English. ⟨pastel-00734697⟩

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