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Journal Articles Numerical Algorithms Year : 2011

Error analysis of Jacobi derivative estimators for noisy signals

Abstract

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.
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Dates and versions

inria-00573270 , version 1 (03-03-2011)

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Da-Yan Liu, Olivier Gibaru, Wilfrid Perruquetti. Error analysis of Jacobi derivative estimators for noisy signals. Numerical Algorithms, 2011, 58 (1), pp.53-83. ⟨10.1007/s11075-011-9447-8⟩. ⟨inria-00573270⟩
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