Investigation of Parameters of the Problem of Spline Approximation of Noisy Data by Numerical Methods of Optimal Control

Abstract

Currently, the problems of distortion of measurement data by noise and the appearance of un-certainties in quality criteria have caused increased interest in research in the field of spline approx-imation. At the same time, existing methods of minimizing empirical risk, assuming that the noise is a uniform distribution with heavier tails than Gaussian, limit the scope of application of these studies. The problem of estimating noise-distorted data is usually based on solving an optimi-zation problem with a function containing uncertainty arising from the problem of finding optimal parameters. In this regard, the estimation of distorted noise cannot be solved by classical methods. Aim. This study is aimed at solving and analyzing the problem of spline approximation of data under uncertainty conditions based on the parametrization of control and the gradient projec-tion algorithm. Methods. The study of the problem of spline approximation of noisy data is carried out by the method of approximation of the piecewise constant control function. In this case, para-metrization of the control is possible only for a finite number of break points of the first kind. In the framework of the experimental study, the gradient projection algorithm is used for the numerical solution of the spline approximation problem. The proposed methods are used to study the parameters of the problem of spline approximation of data under conditions of uncertain-ty. Results. The numerical study of the control parametrization approach and the gradient projec-tion algorithm is based on the developed software and algorithmic tool for solving the problem of the spline approximation model under uncertainty. To evaluate the noise-distorted data, numerical experiments were conducted to study the model parameters and it was found that increasing the value of the parameter α leads to an increase in accuracy, but a loss of smoothness. In addition, the analysis showed that the considered distribution laws did not change the accuracy and convergence rate of the algorithm. Conclusion. The proposed approach for solving the problem of spline approx-imation under uncertainty conditions allows us to determine the problems of distortion of measure-ment data by noise and the appearance of uncertainties in the quality criteria. The study of the model parameters showed that the constructed system is stable to the error of the initial approxima-tion, and the distribution laws do not significantly affect the accuracy and convergence of the gra-dient projection method.