Browsing by Author "Tkachenko, Oleksandr"

Now showing 1 - 7 of 7
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    Application of the Polynomial Maximization Method for Estimating Nonlinear Regression Parameters with Non-Gaussian Asymmetric Errors
    (Springer Nature, 2024) Заболотній, Сергій Васильович; Zabolotnii, Serhii; Tkachenko, Oleksandr; Nowakowski, Waldemar; Warsza, Zygmunt Lech
    In the article, an alternative approach to estimating parameters in nonlinear regression models under asymmetric error distributions is examined. A novel approach for adaptive estimation is proposed, which is based on the use of second-order polynomial functions. This enables a straightforward implementation to account for deviations from Gaussian idealization in the form of moments up to the fourth order. It is demonstrated that the overall problem can algorithmically be reduced to the numerical solution of a system of nonlinear stochastic equations. Analytical expressions are obtained, which facilitate the estimation of parameters and the analysis of their asymptotic variance. Statistical modeling using the Monte Carlo method was conducted, and the results indicate that the accuracy of PMM2 estimates is comparable to SLS estimates and significantly so exceeds the accuracy of OLS estimates.
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    Application of the Polynomial Maximization Method for Estimation Parameters in the Polynomial Regression with Non-Gaussian Advances in Intelligent Systems and Computing
    (Springer Nature, 2021) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Tkachenko, Oleksandr; Ткаченко, Олександр Миколайович; Warsza, Zygmunt Lech
    This paper considers the application of the polynomial maximization method to find estimates of the parameters of polynomial regression. It is shown that this method can be effective for the case when the distribution of the random component of the regression models differs significantly from the Gaussian distribution. This approach is adaptive and is based on the analysis of higher-order statistics of regression residuals. Analytical expressions that allow finding estimates and analyzing their uncertainty are obtained. Cases of asymmetry and symmetry of the distribution of regression errors are considered. It is shown that the variance of estimates of the polynomial maximization method can be significantly less than the variance of the estimates of the least squares method, which is a special case. The increase in accuracy depends on the values of the cumulant coefficients of higher orders of random errors of the regression model. The results of statistical modeling by the Monte Carlo method confirm the effectiveness of the proposed approach.
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    Application of the Polynomial Maximization Method for Estimation Parameters of Autoregressive Models with Asymmetric Innovations
    (Springer Nature, 2022) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Tkachenko, Oleksandr; Warsza, Zygmunt Lech
    This paper considers the application of the Polynomial Maximization Method to find estimates of the parameters of autoregressive model with non-Gaussian innovation. This approach is adaptive and is based on the analysis of higher-order statistics. Analytical expressions that allow finding estimates and analyzing their uncertainty are obtained. Case of asymmetry of the distribution of autoregressive innovations is considered. It is shown that the variance of estimates of the Polynomial Maximization Method can be significantly less than the variance of the estimates of the linear approach (based on Yule-Walker equation or Ordinary Least Squares). The increase in accuracy depends on the values of the cumulant coefficients of higher orders of innovation residuals. The results of statistical modeling by the Monte Carlo method confirm the effectiveness of the proposed approach.
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    Estimating parameters of linear regression with an exponential power distribution of errors by using a polynomial maximization method
    (ПП «ТЕХНОЛОГІЧНИЙ ЦЕНТР», Український державний університет залізничного транспорту, 2021) Заболотній, Сергій Васильович; Zabolotnii, Serhii; Хотунов, Владислав Ігорович; Khotunov, Vladyslav; Чепинога, Анатолій Володимирович; Chepynoha, Anatolii; Ткаченко, Олександр Миколайович; Tkachenko, Oleksandr
    This paper considers the application of a method for maximizing polynomials in order to find estimates of the parameters of a multifactorial linear regression provided the random errors of the regression model follow an exponential power distribution. The method used is conceptually close to a maximum likelihood method because it is based on the maximization of selective statistics in the neighborhood of the true values of the evaluated parameters. However, in contrast to the classical parametric approach, it employs a partial probabilistic description in the form of a limited number of statistics of higher orders. The adaptive algorithm of statistical estimation has been synthesized, which takes into consideration the properties of regression residues and makes it possible to find refined values for the estimates of the parameters of a linear multifactorial regression using the numerical Newton-Rafson iterative procedure. Based on the apparatus of the quantity of extracted information, the analytical expressions have been derived that make it possible to analyze the theoretical accuracy (asymptotic variances) of estimates for the method of maximizing polynomials depending on the magnitude of the exponential power distribution parameters. Statistical modeling was employed to perform a comparative analysis of the variance of estimates obtained using the method of maximizing polynomials with the accuracy of classical methods: the least squares and maximum likelihood. Regions of the greatest efficiency for each studied method have been constructed, depending on the magnitude of the parameter of the form of exponential power distribution and sample size. It has been shown that estimates from the polynomial maximization method may demonstrate a much lower variance compared to the estimates from a least-square method. And, in some cases (for flat-topped distributions and in the absence of a priori information), may exceed the estimates from the maximum likelihood method in terms of accuracy.
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    Estimation of Linear Regression Parameters of Symmetric Non-Gaussian Errors by Polynomial Maximization Method
    (Springer Nature, 2019) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Warsza, Zygmunt Lech; Tkachenko, Oleksandr; Ткаченко, Олександр Миколайович
    In this paper, a new way of estimation of single-factor linear regression parameters of symmetrically distributed non-Gaussian errors is proposed. This new approach is based on the Polynomial Maximization Method (PMM) and uses the description of random variables by higher order statistics (moments and cumulants). Analytic expressions that allow to find estimates and analyze their asymptotic accuracy are obtained for the degree of polynomial S = 3. It is shown that the variance of polynomial estimates can be less than the variance of estimates of the ordinary least squares’ method. The increase of accuracy depends on the values of cumulant coefficients of higher order of the random regression errors. The statistical modeling of the Monte Carlo method has been performed. The results confirm the effectiveness of the proposed approach.
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    Polynomial Estimation of Linear Regression Parameters for the Asymmetric PDF of Errors
    (Springer Nature, 2018) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Warsza, Zygmunt Lech; Tkachenko, Oleksandr; Ткаченко, Олександр Миколайович
    This paper presents a non-standard way of finding estimates of linear regression parameters for the case of asymmetrically distributed errors. This approach is based on the polynomial maximization method (PMM) and uses the moment and cumulant description of random variables. Analytic expressions are obtained that allow one to find estimates and analyze their accuracy for the degree of the polynomial S = 1 and S = 2. It is shown that the variance of polynomial estimates (for S = 2) in the general case is less than the variance of estimates of the ordinary least squares method, which is a particular case of the polynomial maximization method (for S = 1). The increase in accuracy depends on the values of cumulant coefficients of higher orders of random errors of regression. Statistical modeling (Monte Carlo & bootstrapping method) is performed, the results of which confirm the effectiveness of the proposed approach.
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    Polynomial Maximization Method for Estimation Parameters of Asymmetric Non-Gaussian Moving Average Models
    (Springer Nature, 2023) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Tkachenko, Oleksandr; Warsza, Zygmunt Lech
    This paper considers the application of the Polynomial Maximization Method to find estimates of the parameters Non-Gaussian Moving Average model. This approach is adaptive and is based on the analysis of higher-order statistics. Case of asymmetry of the distribution of Moving Average processes is considered. It is shown that the asymptotic variance of estimates of the Polynomial Maximization Method (2nd order) analytical expressions that allow finding estimates and analyzing their uncertainty are obtained. This approach can be significantly less than the variance of the classic estimates based on minimize Conditional Sum of Squares or Maximum Likelihood (in Gaussian case). The increase in accuracy depends on the values of the coefficient’s asymmetry and kurtosis of residuals. The results of statistical modeling by the Monte Carlo Method confirm the effectiveness of the proposed approach.