Ellipsoidal Approximation of the Set of Solutions of Interval Systems of Linear Algebraic Equations
УДК 512.64
Abstract
The article presents the results of the approximation of the set of solutions of interval systems of linear algebraic equations. These systems are used in the problems of modeling linear deterministic processes. It is assumed that the modeled process is described by an output variable and a set of input variables, the measurement errors of which are assumed to be set by known intervals symmetric with respect to the zero value. Traditionally, the sets of solutions of interval systems of linear algebraic equations in applied problems are approximated by a hyper-rectangular whose sides are parallel to the axes of the selected coordinate system. In this paper, we propose to use an ellipsoidal approximation of these sets, which is more efficient. The main results of the work include the substantiation of assumptions about the properties of the modeled process, the choice of a mathematical method for constructing an approximating ellipsoid, the proposed method for forming boundary points, and a numerical method for solving the problem. A computer simulation of the problem of estimating the parameters of a linear process is performed in Excel, which is used for a comparative study of approximations of solutions of interval systems of linear algebraic equations by a hyper-rectangular and an ellipse.
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Канторович Л.В. О некоторых новых подходах к вычислительным методам и обработке наблюдений // Сибирский математический журнал. 1962. Т. 3. № 5.
Шарый С.П. Конечномерный интервальный анализ. Новосибирск, 2017.
Шелудько А.С. Гарантированное оценивание параметров дискретных моделей хаотических процессов // Вестник ЮУрГУ. Серия : Вычислительная математика и информатика. 2018. Т. 7. № 1.
Мадияров М. Н., Оскорбин Н.М., Суханов С.И. Примеры интервального анализа данных в задачах моделирования процессов // Известия Алт. гос. ун-та. 2019. № 1 (105).
Жолен Л. Прикладной интервальный анализ. М. ; Ижевск, 2005.
Zhilin S.I. Simple method for outlier detection in źtting experimental data under interval error // Chemometrics and Intellectual Laboratory Systems. 2007. Vol. 88. № 1.
Singh K., Upadhyaya S. Outlier Detection: Applications And Techniques // International Journal of Computer Science Issues. 2012. Vol. 9. Issue 1. № 3.
Rana P., Pahuja D., Gautam R. A Critical Review on Outlier Detection Techniques // International Journal of Science and Research. 2014. Vol. 3. Issue 12.
Бедринцев А.А., Чепыжов В.В. Двухкритериальная задача построения оптимальных эллипсоидов для представления данных // Информационные технологии и системы - 2014 : сб. трудов. Н. Новгород. 2014.
Бедринцев А.А., Чепыжов В.В., Чернова С.С. Экстремальные эллипсоиды как аппроксиматоры пространства дизайна в задачах предсказательного метамоделирования // Искусственный интеллект и принятие решений. 2015. № 2.
Copyright (c) 2021 Ерлан Канапиянович Ергалиев , Мураткан Набенович Мадияров , Николай Михайлович Оскорбин , Лариса Ленгардовна Смолякова
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