Multivariate numerical differentiation
Type de document :
Compte-rendu et recension critique d'ouvrage
Titre :
Multivariate numerical differentiation
Auteur(s) :
Riachy, Samer [Auteur]
Laboratoire QUARTZ [QUARTZ ]
Non-Asymptotic estimation for online systems [NON-A]
Mboup, Mamadou [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Centre de Recherche en Sciences et Technologies de l'Information et de la Communication - EA 3804 [CRESTIC]
Richard, Jean-Pierre [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Systèmes Non Linéaires et à Retards [SyNeR]
Laboratoire QUARTZ [QUARTZ ]
Non-Asymptotic estimation for online systems [NON-A]
Mboup, Mamadou [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Centre de Recherche en Sciences et Technologies de l'Information et de la Communication - EA 3804 [CRESTIC]
Richard, Jean-Pierre [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Systèmes Non Linéaires et à Retards [SyNeR]
Titre de la revue :
Journal of Computational and Applied Mathematics
Pagination :
1069-1089
Éditeur :
Elsevier
Date de publication :
2011-10-15
ISSN :
0377-0427
Mot(s)-clé(s) en anglais :
operational calculus
finite impulse response filters
Numerical differentiation
multivariable signals
orthogonal polynomials
inverse problems
least squares
finite impulse response filters
Numerical differentiation
multivariable signals
orthogonal polynomials
inverse problems
least squares
Discipline(s) HAL :
Informatique [cs]/Automatique
Résumé en anglais : [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 ...
Lire la suite >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.Lire moins >
Lire la suite >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.Lire moins >
Langue :
Anglais
Vulgarisation :
Non
Collections :
Source :
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