Title :

Error analysis of Jacobi derivative estimators for noisy signals

Author(s) :

Liu, Da-Yan [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Laboratoire d'Automatique, Génie Informatique et Signal [LAGIS]
Gibaru, Olivier [Auteur]
Laboratoire de Métrologie et de Mathématiques Appliquées [L2MA]
Non-Asymptotic estimation for online systems [NON-A]
Perruquetti, Wilfrid [Auteur]
Systèmes Non Linéaires et à Retards [SyNeR]
Non-Asymptotic estimation for online systems [NON-A]

Journal title :

Numerical Algorithms

Pages :

53-83

Publisher :

Springer Verlag

Publication date :

2011-02-22

ISSN :

1017-1398

English keyword(s) :

Numerical differentiation
Jacobi orthogonal polynomials
Stochastic process
Stochastic integrals
Error bound

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]

English abstract : [en]

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, ...
Show more >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.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

• Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189

Source :

Harvested from HAL