Error analysis of Jacobi derivative estimators for noisy signals

Da-Yan Liu 1, 2, 3 Olivier Gibaru 3, 4 Wilfrid Perruquetti 3, 5
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.
Type de document :
Article dans une revue
Numerical Algorithms, Springer Verlag, 2011, 58 (1), pp.53-83. 〈10.1007/s11075-011-9447-8〉
Liste complète des métadonnées

Littérature citée [29 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/inria-00573270
Contributeur : Dayan Liu <>
Soumis le : jeudi 3 mars 2011 - 11:45:04
Dernière modification le : mardi 3 juillet 2018 - 11:47:31
Document(s) archivé(s) le : samedi 4 juin 2011 - 02:45:57

Fichiers

finale_corrige_V1.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

Collections

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〉

Partager

Métriques

Consultations de la notice

506

Téléchargements de fichiers

349