Browsing by Author "Warsza, Zygmunt Lech"

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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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    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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    Estimation of Measurand Parameters for Data from Asymmetric Distributions by Polynomial Maximization Method
    (Springer Nature, 2018) Warsza, Zygmunt Lech; Zabolotnii, Serhii; Заболотній, Сергій Васильович
    In 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.
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    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 Васильович
    Przedstawiono 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.
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    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 Васильович
    The 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.
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    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.
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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 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 Васильович
    This 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.
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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.