Multivariate numerical differentiation

  • Authors:

  • Samer Riachy;Mamadou Mboup;Jean-Pierre Richard

  • Affiliations:

  • Ecole National Supérieure de l'Electronique et de ses Applications, ECS-Lab/EA-3649, 6, avenue du Ponceau, 95014, Cergy-Pontoise, France;Université de Reims Champagne Ardenne, CReSTIC-UFR SEN, BP 1039 Moulin de la Housse, F-51687 Reims cedex 2, France;Université Lille Nord de France, ECLille, LAGIS, UMR 8146 CNRS, Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve d'Ascq, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2011

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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 lead 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 spatial 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.