Abstract
Recent algebraic parametric estimation techniques (see Fliess and Sira-Ramírez, ESAIM Control Optim Calc Variat 9:151–168, 2003, 2008) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see Mboup et al. 2007, Numer Algorithms 50(4):439–467, 2009). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, 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:
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(a)
the bias error term, due to the truncation, can be reduced by tuning the parameters,
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(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),
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(c)
the variance of the noise error is shown to be smaller in the case of negative real parameters than it was in Mboup et al. (2007, Numer Algorithms 50(4):439–467, 2009) for integer values.
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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Liu, DY., Gibaru, O. & Perruquetti, W. Error analysis of Jacobi derivative estimators for noisy signals. Numer Algor 58, 53–83 (2011). https://doi.org/10.1007/s11075-011-9447-8
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DOI: https://doi.org/10.1007/s11075-011-9447-8
Keywords
- Numerical differentiation
- Jacobi orthogonal polynomials
- Stochastic process
- Stochastic integrals
- Error bound