Multivariate numerical differentiation
Multivariate numerical differentiation
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DOI：
10.1016/j.cam.2011.07.031
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发表时间：
2011-10
期刊：
影响因子：
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通讯作者：
S. Riachy;M. Mboup;J. Richard
中科院分区：
文献类型：
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作者：
S. Riachy;M. Mboup;J. Richard
本文提出了一种新的多元数值微分方法，即多维噪声信号偏导数的估计。从一个本地模型的信号组成的截断泰勒展开，我们表示，通过适当的微分代数操作，所需的偏导数作为一个函数的噪声信号的迭代积分。迭代积分提供噪声过滤。所提出的方法导致家庭的估计每个偏导数的任何顺序。我们提出了一个详细的研究中给出的一些结构性质的递归关系的元素之间的同一家庭。这些属性接下来用于研究估计器的性能。我们发现，一些微分代数操作对应于一个特定的家庭的估计导致隐含的正交投影所需的衍生物的Jacobi多项式基函数，产生的解释流行的最小二乘。这种解释允许一个（1）解释存在的空间延迟固有的估计和（2）推导出一个明确的公式的延迟。我们还展示了如何可以设计，由一个给定的阶导数的不同的基本估计的适当组合，估计给出任何规定的值的延迟。仿真结果表明，无时延估计器对噪声敏感。相对于噪声的鲁棒性可以大大增加，利用任意延迟估计。给出了一种有限冲激响应数字滤波器的数值实现方案。我们的衍生估计的有效性证明了几个数值模拟。
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.