Browsing by Author "Zabolotnii, Serhii"
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Item Application of the matrix factor analysis method for determining parameters of the objective function for transport risk minimization(Lublin University of Technology, 2021) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Mogilei, Sergii; Могілей Сергій ОлександровичThe paper regards a common transport problem with a non-classic optimization criterion to minimize transportation risks. It demonstratesthat the risk parameters of the function could be found through the factor analysis method. Besides, considering that the problem contains several pointsof sending and delivering loads, the method is dealt with as a matrix. The research also regards the algorithm of matrix factor analysis applicationfor determining parameters of the objective function for the problem to be solved. The survey results in a new method to construct the objective functionfor the optimization problem with probability parameters. It generally assists in suggesting a formal solution to such problems, foremost due to particular software.Item 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 LechIn 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.Item 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 LechThis 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.Item 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 LechThis 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.Item Constructing Reference Plans of Two-Criteria Multimodal Transport Problem(Transport & Telecommunication Institute, Latvia, 2021) Przystupa, Krzysztof; Qin, Zhang; Zabolotnii, Serhii; Заболотній, Сергій Васильович; Pohrebennyk, Volodymyr; Mogilei, Sergii; Могілей, Сергій Олександрович; Zhongju, Chen; Gil, LeszekThe object of this study is a multicriteria transport problem, being stated for availability of several means of cargo delivery, meaning a multimodal transport problem. The optimization criteria of the multimodal transport problem described above are two objective functions of minimizing total transportation costs and level of transport risks. Three types of transport were selected for research: automobile, rail and river (inland waterway). The results of the study lay the foundation for development of a new valid algorithm for solving multimodal transport problems like multi-criteria optimization ones. The main advantage of such an algorithm lies in its higher potential convergence rate compared to classical numerical optimization methods, which now are predominantly used to solve the problems of this type. This advantage may not be decisive, but it appears to be at least quite an important argument when choosing the method of realization for two-criteria multimodal transport problems earlier considered, especially, in case of a large dimension. Moreover, the algorithm described in the work can be applied to similar problems with any number of types of transport and optimization criteria.Item 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, OleksandrThis 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.Item 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 МиколайовичIn 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.Item Estimation of Measurand Parameters for Data from Asymmetric Distributions by Polynomial Maximization Method(Springer Nature, 2018) Warsza, Zygmunt Lech; Zabolotnii, Serhii; Заболотній, Сергій ВасильовичIn this paper the non-conventional method for evaluating the standard uncertainty of the estimator of measurand value obtained from the non-Gaussian asymmetrically distributed sampled data with a priori partial description (known only few initial moments, unknown PDF distribution of population) is proposed. This method of statistical estimation is based on the apparatus of maximization the stochastic polynomials (PMM method proposed by Kunchenko) and uses the higher-order statistics (moment or cumulant description) of random variables. The analytical expressions for finding estimates and analyzing their accuracy to the degree of the polynomial s = 2 is given. It is shown that for the asymmetric PDF-s the uncertainty estimates for received polynomial are generally smaller than the uncertainty estimates obtained based on the mean (arithmetic average). Reducing the uncertainty of measurement depends on the skewness and kurtosis. On the basis of the Monte Carlo method statistical modelling is carried out, the results confirm the effectiveness of the proposed approach.Item Estimation of measurand parameters for data from asymmetric distributions by polynomial maximization method (PMM)(Industrial Research Institute for Automation and Measurements PIAP, 2018) Warsza, Zygmunt Lech; Zabolotnii, Serhii; Заболотній, Сергій ВасильовичPrzedstawiono sposób wyznaczania estymatorów wartości i niepewności menzurandu niekonwencjonalną metodą maksymalizacji wielomianu stochastycznego (PMM) dla próbki danych pomiarowych pobranych z populacji modelowanej zmienną losową o rozkładzie niesymetrycznym. W metodzie PMM stosuje się statystykę wyższego rzędu i opis z użyciem momentów lub kumulantów. Wyznaczono wyrażenia analityczne dla estymatorów wartości i niepewności standardowej typu A menzurandu za pomocą wielomianu stopnia r = 2. Niepewność standardowa wartości menzurandu otrzymana metodą PPM zależy od skośności i kurtozy rozkładu. Jest ona mniejsza od średniej arytmetycznej wyznaczanej wg przewodnika GUM i bliższa wartości teoretycznej dla rozkładu populacji danych. Jeśli rozkład ten jest nieznany, to estymatory momentów i kumulantów wyznacza się z danych pomiarowych próbki. Sprawdzono skuteczność metody PMM dla kilku podstawowych rozkładów.Item Factor analysis method application for constructing objective functions of optimization in multimodal transport problems(Lublin University of Technology, 2021) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Honcharov, Artem; Гончаров, Артем Володимирович; Mogilei, Sergii; Могілей, Сергій ОлександровичThe paper regards a specific class of optimization criteria that possess features of probability. Therefore, constructing objective function of optimization problem, the importance is attached to probability indices that show the probability of some criterial event or events to occur. Factor analysis has been taken for the main method of constructing objective function. Algorithm for constructing objective function of optimization is donefor criterion of minimization risk level in multimodaltransportations that demanded demonstration data. The application of factor analysis in classical problem solution was shown to givethe problem a more distinct analytical interpretation in solving it.Item Method of Verification of Hypothesis about Mean Value on a Basis of Expansion in a Space with Generating Element(Allerton Press, Inc., 2018) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Martynenko, S. S.; Salypa, S. V.In this paper it is proposed an original method for verification of statistical hypotheses about mean values of random quantities. This method is based on Kunchenko stochastic polynomials tool and probabilistic description on a basis of higher order statistics (moments and/or cumulants). There are represented analytical expressions allowing to optimize decision rules using certain qualitive criterion and calculate decision-making error. It is shown polynomial decision rule in case of polynomial power S = 1 corresponds to classic linear decision rule which is used for comparative analysis. By means of multiple statistical experiments (Monte–Carlo method) obtained results of Neumann–Pierson criterion show proposed polynomial decision rules are characterized by increased accuracy (decrease of the 2nd genus errors probability) in compare to linear processing. The method efficiency increases with increase of stochastic polynomial order increase of degree of random quantities distribution difference from Gaussian probabilities distribution law.Item Modifications of Evans Price Equilibrium Model(Lublin University of Technology, 2023) Заболотній, Сергій Васильович; Zabolotnii, Serhii; Могілей, Сергій Олександрович; Mogilei, SergiiThe paper regards the classical Evans price equilibrium model in the free product market in the aspect of regarding the opportunitiesfor expanding (modifying) the model given that is aimed at perfecting the accuracy of its mathematical formulating. As an accuracy criterion, we have chosen a summary quadratic deviation of the calculated indices from the given ones. One of the approaches of modifying the basic Evans modelis suggesting there is a linear dependence between price function and time as well as its first and second derivatives. In this case, the model willbe described through differential equation of second order with constant coefficients,revealing some oscillatory process. Besides, it is worth regardinga non-linear (polynomial) dependence between demand, supply and price.The paper proposes mathematical formulating for the modified Evans models that have been approbated for real indices of exchange rates fluctuations. It also proves that increase of the differential and/or polynomial orderof the given model allows its essential accuracy perfection. Besides, the influence of arbitrary restricting circumstances ofthe model onits accuracyis regarded. Each expanded Evans model is accompanied by mathematically formulated price and time dependence.Item Ocena niepewności pomiarów o rozkładzie trapezowym metodą maksymalizacji wielomianu(Wydawnictwo Czasopism i Książek Technicznych SIGMA-NOT Sp. z o.o., 2017) Warsza, Zygmunt Lech; Zabolotnii, Serhii; Заболотній, Сергій ВасильовичThe types of measurand parameter estimators derived from samples of measured data taken from a sym. trapezoidal population were briefly reviewed (9 refs.). A non-std. approach to find ests. of the non-Gaussian distributions parameters based on the unconventional method for maximizing the stochastic polynomials by using a moment-cumulant description of random variables was proposed. The method was recommended to use for detg. estd. values of the std. deviation and uncertainties of measurand when distribution of the random errors population is a priori unknown and first few cumulants have to be found from the sample data. The method is particularly useful in assessing mixts. and mixing efficiency.Item Optimization of the method of constructing reference plans of multimodal transport problem(ПП «Технологічний центр», Полтавська державна аграрна академія, 2018) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Mogilei, Sergii; Могілей, Сергій ОлександровичКласична транспортна задача полягає у визначенні оптимального плану перевезень вантажів з пунктів відправки до пунктів доставки за критерієм мінімальної собівартості таких перевезень. Така задача враховує лише один вид транспорту, що в недостатній мірі відповідає практичним потребам сучасних логістичних підприємств. Саме тому об’єктом даного дослідження є класична транспортна задача, по-становка якої враховує наявність кількох засобів доставки вантажу, а саме: автомобільного, залізничного та водного. Транспортну задачу такого типу визначено як мультимодальну. Реалізація мультимодальної транспортної задачі передбачає використання різноманітних чисельних методів та виконується за допомогою програмних засобів. Фактично, концептуальний підхід до її розв’я-зання полягає в простому підборі можливих розв’язків. За умови великої розмірності задачі такий підхід може бути надзвичайно громіздким, а тому потребує певного удосконалення. Під час проведення дослідження було оптимізовано метод побудови опорного плану такої задачі на основі критерію мінімізації кількості чисельних ітерацій, обґрунтовано переваги запропонованого підходу у порівнянні з уже відомими. В основу нового підходу було покладено раніше відомий метод мінімального елемента, що використовується при розв’язанні транспортної задачі, а також проведе-но аналогію із задачею Штейнера. Останнє, в свою чергу, дало змогу означити новий підхід як метод Штейнера. Результатом дослідження є розробка загального алгоритму реалізації запропонованого методу Штейнера. В якості апробації даного алгоритму подано модельний приклад, який демонструє ідентичність результатів розв’язання мультимодальної транспортної задачі всіма розглянутими в роботі способами. Розробка нових методів реалізації мультимодальної транспортної задачі дозволить побудувати ефек-тивні алгоритми розв’язання більш комплексних задач транспортної логістики. Критерій зменшення кількості чисельних ітерацій, застосований на всіх етапах реалізації таких задач, значно скоротить час відшукання їхніх розв’язків.Item Polynomial Estimates of Measurand Parameters for Data from Bimodal Mixtures of Exponential Distributions(Karagandy University of the name of academician E.A. Buketov, 2018) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Kucheruk, V. Yu.; Warsza, Zygmunt Lech; Khassenov, А. K.A non-conventional approach to finding estimates of the result of multiple measurements for a random error model in the form of bimodal mixtures of exponential distributions is proposed. This approach is based on the application of the Polynomial Maximization Method (PMM) with the description of random variables by higher order statistics (moment & cumulant). The analytical expressions for finding estimates and analysis accuracy to the degree of the polynomial r = 3 are presented. In case when the degree of the polynomial r = 1 and r = 2 (for symmetrically distributed data) polynomial estimate equivalent can be estimated as a mean (average arithmetic). In case when the degree of the polynomial r = 3, the uncertainty of the polynomial estimate decreases. The reduction coefficient depends on the values of the 4th and 6th order cumulant coefficients that characterize the degree of difference while the distribution of sample data from the Gaussian model. By means of multiple statistical tests (Monte Carlo method), the properties of the normalization of polynomial estimates are investigated and a comparative analysis of their accuracy with known estimates (mean, median and center of folds) is made. Areas that depend on the depth of antimodality and sample size, in which polynomial estimates (for r = 3) are the most effective.Item 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 МиколайовичThis 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.Item Polynomial estimation of the measurand parameters for samples from non-Gaussian distributions based on higher order statistics(World Scientific Publishing Co Pte Ltd, 2019) Warsza, Zygmunt Lech; Zabolotnii, Serhii; Заболотній, Сергій ВасильовичThis paper proposes an unconventional method (PMM) for evaluating the uncertainty of the estimator of measurand value obtained from the non-Gaussian distributed samples of measurement data with a priori partial description (unknown PDF). This method of statistical estimation is based on the apparatus of stochastic polynomial maximization and uses the higher-order statistics (moment and cumulant description) of random variables. The analytical expressions for estimates of uncertainty, obtained with use the polynomial of the degree r = 2 for samples from population of asymmetrical pdf and degree r = 3 — for symmetrical pdf, are given. It is shown that these uncertainties are generally smaller than the uncertainty based only on the arithmetic average, as it is in GUM. Reducing the value of estimated uncertainty of measurement depends on the skewness and kurtosis of samples from asymmetrical pdf or on kurtosis and six order moment of samples from symmetrical pdf. The results of statistical modeling carried out on the basis of the Monte Carlo method confirm the effectiveness of the proposed approach.Item 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 LechThis 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.Item Сomparative Analysis of Polynomial Maximization and Maximum Likelihood Estimates for Data with Exponential Power Distribution(Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», 2020) Zabolotnii, Serhii; Заболотній, Сергій Васильович; Chepynoha, Anatolii; Чепинога, Анатолій Володимирович; Chorniy, Andriy; Чорній, Андрій Михайлович; Honcharov, Artem; Гончаров, Артем ВолодимировичThe work is devoted to the estimate accuracy comparative analysis of the experimental data parameters with exponential power distribution (EPD) using the classical Maximum Likelihood Estimation (MLE) and the original Polynomial Maximization Method (PMM). In contrast to the parametric approach of MLE, which uses the description in the form of probability density distribution, PMM is based on a partial description in the of higher-order statistics form and the mathematical apparatus of Kunchenko's stochastic polynomials. An algorithm for finding PMM estimates using 3rd order stochastic polynomials is presented. Analytical expressions allowing to determine the variance of PMM-estimates of the asymptotic case parameters and EPD parameters with a priori information are obtained. It is shown that the relative theoretical estimates accuracy of different methods significantly depends on the EPD shape parameter and matches only for a separate case of Gaussian distribution. The effectiveness of different approaches (including valuation of mean values estimates) both with and without a priori information on EPD properties was investigated by repeated statistical tests (through Monte Carlo Method). The greatest efficiency areas for each of methods depending on EPD shape parameter and sample data volume are constructed.