### METHOD FOR NUMERICAL DIFFERENTIATION OF NOISY DATA WITH OUTLIERS

#### Abstract

Context. Using of conventional methods of numerical differentiation to the noisy data with outliers leads to significant errors. The objectof this study is the process of numerical differentiation of such data.

Objective. The goal of this work is the development of a method of numerical differentiation of the noisy data with outliers to obtain a smooth approximation of the first derivative of original data as well as a smooth approximation of the original data themselves.

Method. The proposed method of numerical differentiation is based on solving the problem of minimizing the smoothing functional, which is built on the criteria of a minimum of extent of the solution residual and of an energy constraint of the first derivative of solution. The minimum-extent criterion defines the main part of functional and ensures its stable behavior with respect to the additive noise and outliers. The energy constraint defines the stabilizing part of the functional and provides a smooth solution of the problem. The contribution of these parts is controlled by a regularization parameter. Since the main part of smoothing functional is not convex, then the minimization problem is the non-convex nonlinear programming problem. For the numerical solution of this problem the conjugate gradient method is used. In this method the step size along the descent direction is defined on the set of test steps. These steps minimize the individual components of the main and stabilizing parts of the smoothing functional that allows to move from the one local minimum of the functional to another deeper local minimum.

Results. Simulation of the problem of numerical differentiation of noisy data with outliers and processing of the experimental data, which are photoluminescence spectra with narrow line components in their compositions, confirmed the performance of the proposed method.

Conclusions. The proposed method can be used for numerical differentiation of noisy data with outliers. It provides a smooth approximation of the first derivative of the original data, as well as a smooth approximation of the original data themselves. This method can be generalized to the case of non-smooth solutions by constructing a stabilizing part of the functional based on the criterion of minimum total variation.

#### Keywords

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#### GOST Style Citations

1. Derivative spectroscopy and the continuous relaxation spectrum / [A. R. Davies, R. S. Anderssen, F. R. de Hoog, N. J. Goulding] // Journal of Non-Newtonian Fluid Mechanics. – 2016. – Vol. 233, July. – P. 107–118. DOI: 10.1016/j.jnnfm.2016.01.018.

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4. Selected algorithms for measurement data processing in impulseradar-based system for monitoring of human movements / [A. Miкkina, J. Wagner, P. Mazurek, R. Z. Morawski] // Journal of Physics. – 2016. – Vol. Conference Series 772. – P. 1–6. DOI:10.1088/1742-6596/772//012057.

5. Knowles I. Methods for numerical differentiation of noisy data / I. Knowles, R. J. Renka // Electronic Journal of Differential Equations. – 2014. – Vol. Conference 21. – P. 235–246.

6. Chartrand R. Numerical differentiation of noisy, nonsmooth data / R. Chartrand // ISRN Applied Mathematics. – 2011. – Vol. 2011. – P. 1–11. DOI: 10.5402/2011/164564.

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9. Вовк С. М. Двойственный метод минимума пространственной протяженности для робастного оценивания параметров дипольных источников излучения / С. М. Вовк, В. Ф. Борулько // Радіоелектроніка, інформатика, управління. – 2014. – № 2. – С. 8–17. DOI: 10.15588/1607-3274-2014-2-1.

10. Vovk S. M. General approach to building the methods of filtering based on the minimum duration principle / S. M. Vovk // Radioelectronics and Communications Systems. – 2016. – Vol. 59. – № 7. – P. 281–292. DOI: 10.3103/S0735272716070013.

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DOI: https://doi.org/10.15588/1607-3274-2017-3-5

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