Multivariate numerical differentiation

Samer Riachy 1, 2 Mamadou Mboup 2, 3 Jean-Pierre Richard 2, 4, 5
2 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
4 SyNeR - Systèmes Non Linéaires et à Retards
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : 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.
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Journal articles
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https://hal.inria.fr/inria-00637164
Contributor : Samer Riachy <>
Submitted on : Monday, October 31, 2011 - 7:15:02 AM
Last modification on : Monday, September 2, 2019 - 2:42:14 PM
Long-term archiving on: Wednesday, February 1, 2012 - 2:20:35 AM

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Samer Riachy, Mamadou Mboup, Jean-Pierre Richard. Multivariate numerical differentiation. JCAM, Elsevier, 2011. ⟨inria-00637164⟩

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