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Journal of Radio Electronics Online scientific journal · eISSN 1684-1719
2026, № 9 · RADIO ENGINEERING

Statistical linearization and cumulant method in the analysis of non-linear stochastic systems

V.M. Artyushenko1, V.I. Volovach2,3

1 Moscow State University of Geodesy and Cartography, Moscow, Russia
2 Volga Region State University of Service, Togliatti, Russia
3 MIREA – Russian Technological University, Moscow, Russia
PDF DOI: 10.30898/1684-1719.2026.9.3 Published 8 November 2026

Abstract

The article examines the methods of analysis of non-linear stochastic systems based on statistical linearization and the apparatus of cumulants. Two approaches to determining statistical linearization coefficients based on criteria of coincidence of statistical characteristics and minimization of mean square error of approximation are considered. Analytical expressions were obtained for the parameters of statistically linearized models, as well as the features of their application in the analysis of non-Gaussian random processes were investigated. Various statistical approximation methods were compared using a non-linear dynamic system under the influence of white Gaussian noise. The analysis of the cumulants method was carried out; the features of using the Edgewart decomposition to construct a closed system of differential equations relative to the first cumulants of a random process were considered. It is shown that the statistical linearization method is a special case of the cumulant method with the Gaussian approximation of the probability distribution. A comparative assessment of the accuracy of the methods considered was carried out, which makes it possible to determine the areas of their effective application in the study of nonlinear stochastic systems.

  • statistical linearization
  • non-linear stochastic system
  • cumulant method
  • semiinvariant
  • non-Gaussian random process
  • Edgeworth series
  • stochastic differential equation
  • correlation characteristics
Аннотация (на русском)

Статистическая линеаризация и метод кумулянтов при анализе нелинейных стохастических систем

В статье исследованы методы анализа нелинейных стохастических систем, основанные на статистической линеаризации и аппарате кумулянтов. Рассмотрены два подхода к определению коэффициентов статистической линеаризации, основанные на критериях совпадения статистических характеристик и минимизации среднеквадратической ошибки аппроксимации. Получены аналитические выражения для параметров статистически линеаризованных моделей, а также исследованы особенности их применения при анализе негауссовских случайных процессов. Выполнено сравнение различных способов статистической аппроксимации на примере нелинейной динамической системы, находящейся под воздействием белого гауссовского шума. Проведен анализ метода кумулянтов, рассмотрены особенности использования разложения Эджварта для построения замкнутой системы дифференциальных уравнений относительно первых кумулянтов случайного процесса. Показано, что метод статистической линеаризации представляет собой частный случай метода кумулянтов при гауссовском приближении распределения вероятностей. Проведена сравнительная оценка точности рассмотренных методов, позволяющая определить области их эффективного применения при исследовании нелинейных стохастических систем.

Funding

The work was carried out at the expense of the State Assignment №FSFZ-2024-0006.

For citation

Artyushenko V.M., Volovach V.I. Statistical linearization and cumulant method in the analysis of non-linear stochastic systems // Journal of Radio Electronics. – 2026. – No. 9. https://doi.org/10.30898/1684-1719.2026.9.3

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