Title :

Multivariate numerical differentiation

Author(s) :

Riachy, Samer [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Laboratoire QUARTZ [QUARTZ ]
Mboup, Mamadou [Auteur]
Centre de Recherche en Sciences et Technologies de l'Information et de la Communication - EA 3804 [CRESTIC]
Non-Asymptotic estimation for online systems [NON-A]
Richard, Jean-Pierre [Auteur]
Systèmes Non Linéaires et à Retards [SyNeR]
Non-Asymptotic estimation for online systems [NON-A]

Journal title :

Journal of Computational and Applied Mathematics

Pages :

1069-1089

Publisher :

Elsevier

Publication date :

2011-10-15

ISSN :

0377-0427

English keyword(s) :

operational calculus
finite impulse response filters
Numerical differentiation
multivariable signals
orthogonal polynomials
inverse problems
least squares

HAL domain(s) :

Informatique [cs]/Automatique

English abstract : [en]

We present an innovative method for multivariate numerical differentiation i.e. the estimation of partial derivatives of multidimensional noisy signals. Starting from a local model of the signal consisting of a truncated ...
Show more >We present an innovative method for multivariate numerical differentiation i.e. the estimation of partial derivatives of multidimensional noisy signals. Starting from a local model of the signal consisting of a truncated Taylor expansion, we express, through adequate differential algebraic manipulations, the desired partial derivative as a function of iterated integrals of the noisy signal. Iterated integrals provide noise filtering. The presented method leads to a family of estimators for each partial derivative of any order. We present a detailed study of some structural properties given in terms of recurrence relations between elements of a same family. These properties are next used to study the performance of the estimators. We show that some differential algebraic manipulations corresponding to a particular family of estimators leads implicitly to an orthogonal projection of the desired derivative in a Jacobi polynomial basis functions, yielding an interpretation in terms of the popular least squares. This interpretation allows one to 1) explain the presence of a spacial delay inherent to the estimators and 2) derive an explicit formula for the delay. We also show how one can devise, by a proper combination of different elementary estimators of a given order derivative, an estimator giving a delay of any prescribed value. The simulation results show that delay-free estimators are sensitive to noise. Robustness with respect to noise can be highly increased by utilizing voluntary-delayed estimators. A numerical implementation scheme is given in the form of finite impulse response digital filters. The effectiveness of our derivative estimators is attested by several numerical simulations.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