# Error analysis of Jacobi derivative estimators for noisy signals

3 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
5 SyNeR - Systèmes Non Linéaires et à Retards
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : Recent algebraic parametric estimation techniques (see \cite{garnier,mfhsr}) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see \cite{num0,num}). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, %as in \cite{num0,num}, the extension is done via a truncated Jacobi orthogonal series expansion. Then, the noise error contribution of these derivative estimations is investigated: after proving the existence of such integral with a stochastic process noise, their statistical properties (mean value, variance and covariance) are analyzed. In particular, the following important results are obtained: \begin{description} \item[$a)$] the bias error term, due to the truncation, can be reduced by tuning the parameters, \item[$b)$] such estimators can cope with a large class of noises for which the mean and covariance are polynomials in time (with degree smaller than the order of derivative to be estimated), \item[$c)$] the variance of the noise error is shown to be smaller in the case of negative real parameters than it was in \cite{num0,num} for integer values. \end{description} Consequently, these derivative estimations can be improved by tuning the parameters according to the here obtained knowledge of the parameters' influence on the error bounds.
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Journal articles

Cited literature [29 references]

https://hal.inria.fr/inria-00573270
Contributor : Dayan Liu <>
Submitted on : Thursday, March 3, 2011 - 11:45:04 AM
Last modification on : Thursday, October 1, 2020 - 12:48:07 PM
Long-term archiving on: : Saturday, June 4, 2011 - 2:45:57 AM

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### Citation

Da-Yan Liu, Olivier Gibaru, Wilfrid Perruquetti. Error analysis of Jacobi derivative estimators for noisy signals. Numerical Algorithms, Springer Verlag, 2011, 58 (1), pp.53-83. ⟨10.1007/s11075-011-9447-8⟩. ⟨inria-00573270⟩

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