186 research outputs found
Mathematical model of the flat problem of the allowance distribution
Рассматривается плоская проблема распределения припуска. Предлагается и обсуждается алгоритм компьютерного расчета припуска в условиях поточечного задания контуров заготовки и изделия
Dual control of multidimensional-matrix stochastic objects
The main results in dual control theory are analyzed. The problem of the dual control of the multidimensional-
matrix stochastic objects is formulated and the functional equations for its solution are given
Дуальная стабилизация многомерного регрессионного объекта на заданном уровне
The statement of the problem of the dual control of the regression object with multidimensional-matrix
input and output variables and dynamic programming functional equations for its solution are given. The problem
of the dual stabilization of the regression object at the given level is considered. The purpose of control is reaching
the given value of the output variable by sequential control actions in production operation mode. In order to solve
the problem, the regression function of the object is supposed to be affine in input variables, and the inner noise
is supposed to be Gaussian. The sequential solution of the functional dynamic programming equations is performed. As a result, the optimal control action at the last control step is obtained. It is shown also that the obtaining
of the optimal control actions at the other control steps is connected with big difficulties and impossible both analytically and numerically. The control action obtained at the last control step is proposed to be used at the arbitrary
control step. This control action is called the control action with passive information accumulation. The dual
control algorithm with passive information accumulation was programmed for numerical calculations and tested
for a number of objects. It showed acceptable results for the practice. The advantages of the developed algorithm
are theoretical and algorithmical generality
Total probability and bayes formulae for joint multidimensional–matrix Gaussian distributions
This paper is devoted to the development of a mathematical tool for obtaining the Bayesian estimations of the parameters of multidimensional regression objects in their finite-dimensional multidimensional-matrix description. Such a need arises, particularly, in the problem of dual control of regression objects when multidimensional-matrix mathematical formalism is used for the description of the controlled object. In this paper, the concept of a one-dimensional random cell is introduced as a set of multidimensional random matrices (in accordance with the “cell array” data type in the Matlab programming system), and the definition of the joint multidimensional-matrix Gaussian distribution is given (the definition of the Gaussian one-dimensional random cell). This required the introduction of the concepts of one-dimensional cell of the mathematical expectation and two-dimensional cell of the variance-covariance of the one-dimensional random cell. The integral connected with the joint Gaussian probability density function of the multidimensional matrices is calculated. The two formulae of the total probability and the Bayes formula for joint multidimensional-matrix Gaussian distributions are given. Using these results, the Bayesian estimations of the unknown coefficients of the multidimensional-matrix polynomial regression function are obtained. The algorithm of the calculation of the Bayesian estimations is realized in the form of the computer program. The results represented in the paper have theoretical and algorithmic generality
On the dual stabilization of the multidimensional regression object at a given level
The statement of the problem of the dual control of the regression object with multidimensional-matrix input and output variables and dynamic programming functional equations for its solution are given. The problem of the dual stabilization of the regression object at the given level is considered. In order to solve the problem, the regression function of the object is supposed to be affine in input variables, and the inner noise is supposed to be Gaussian. The optimal control action at the last control step is obtained and is proposed to be used at the arbitrary control step. The obtained control algorithm was programmed for numerical calculations and tested for a number of objects
Dual Control of the Extremal Multidimensional Regression Object
The statement of the problem of the dual control of the regression object with multidimensional-
matrix input and output variables and dynamic programming functional equations for its solution are given.
The problem of the dual control of the extremal regression object, i.e. object response function of which has an
extremum, is considered. The purpose of control is reaching the extremum of the output variable by sequential
control actions in production operation mode. In order to solve the problem, the regression function of the
object is supposed to be quadratic in input variables, and the inner noise is supposed to be Gaussian. The
sequential solution of the functional dynamic programming equations is performed. As a result, the optimal
control action at the last control step is obtained. It is shoved also that the optimal control actions obtaining at
the other control steps is connected with big difficulties and impossible both analytically and numerically. The
control action obtained at the last control step is proposed to be used at the arbitrary control step. This control
action is called the control action with passive information accumulation. The dual control algorithm with
passive information accumulation was programmed for numerical calculations and tested for a number of
objects. It showed acceptable results for the practice
Формулы полной вероятности и Байеса для совместных многомерно-матричных гауссовских распределений
This paper is devoted to the development of a mathematical tool for obtaining the Bayesian estimations of the parameters of multidimensional regression objects in their finite-dimensional multidimensional-matrix description. Such a need arises, particularly, in the problem of dual control of regression objects when multidimensional-matrix mathematical formalism is used for the description of the controlled object. In this paper, the concept of a one-dimensional random cell is introduced as a set of multidimensional random matrices (in accordance with the “cell array” data type in the Matlab programming system), and the definition of the joint multidimensional-matrix Gaussian distribution is given (the definition of the Gaussian one-dimensional random cell). This required the introduction of the concepts of one-dimensional cell of the mathematical expectation and two-dimensional cell of the variance-covariance of the one-dimensional random cell. The integral connected with the joint Gaussian probability density function of the multidimensional matrices is calculated. The two formulae of the total probability and the Bayes formula for joint multidimensional-matrix Gaussian distributions are given. Using these results, the Bayesian estimations of the unknown coefficients of the multidimensional-matrix polynomial regression function are obtained. The algorithm of the calculation of the Bayesian estimations is realized in the form of the computer program. The results represented in the paper have theoretical and algorithmic generality.Работа посвящена разработке математического аппарата для получения байесовских оценок параметров многомерных регрессионных объектов в их конечномерном многомерно-матричном описании. Такая потребность возникает, в частности, в задаче дуального управления регрессионными объектами, когда для описания многомерного управляемого объекта применяется многомерно-матричный математический аппарат. В статье вводится понятие одномерной случайной ячейки как совокупности многомерных случайных матриц (в соответствии с данными типа «массив ячеек» в системе программирования Матлаб) и дается определение совместного гауссовского распределения многомерных случайных матриц (определение гауссовской одномерной случайной ячейки). Это потребовало введения понятия одномерной ячейки математического ожидания и понятия двумерной ячейки вариаций-ковариаций одномерной случайной ячейки. Далее вычисляется один интеграл, связанный с функцией совместной гауссовской плотности вероятности многомерных случайных матриц. Приводятся две формулы полной вероятности и формула Байеса для совместных многомерно-матричных гауссовских распределений. На основе этих результатов получены байесовские оценки неизвестных коэффициентов многомерно-матричной полиномиальной функции регрессии. Алгоритм расчета байесовских оценок реализован в виде компьютерной программы. Представленные результаты обладают теоретической и алгоритмической общностью
Формулы полной вероятности и Байеса для совместных многомерно-матричных гауссовских распределений
This paper is devoted to the development of a mathematical tool for obtaining the Bayesian estimations of
the parameters of multidimensional regression objects in their finite-dimensional multidimensional-matrix description. Such a need arises, particularly, in the problem of dual control of regression objects when multidimensional-matrix mathematical formalism is used for the description of the controlled object. In this paper, the concept of a one-dimensional random cell is introduced as a set of multidimensional random matrices (in accordance with the “cell array” data type in the Matlab programming system), and the definition of the joint multidimensional-matrix Gaussian distribution is given (the definition of the Gaussian one-dimensional random cell). This required the introduction of the concepts of one-dimensional cell of the mathematical expectation and two-dimensional cell of the variance-covariance of the one-dimensional random cell. The integral connected with the joint Gaussian probability density function of the multidimensional matrices is calculated. The two formulae of the total probability and the Bayes formula for joint multidimensional-matrix Gaussian distributions are given. Using these results, the Bayesian estimations of the unknown coefficients of the multidimensional-matrix polynomial regression function are obtained. The algorithm of the calculation of the Bayesian estimations is realized in the form of the computer program. The results represented in the paper have theoretical and algorithmic generality
Multidimensional regression object at a given level
The statement of the problem of the dual control of the regression object with multidimensional-matrix input and output variables and dynamic programming functional equations for its solution are given. The problem of the dual stabilization of the regression object at the given level is considered. In order to solve the problem, the regression function of the object is supposed to be affine in input variables, and the inner noise is supposed to be Gaussian. The optimal control action at the last control step is obtained and is proposed to be used at the arbitrary control step. The obtained control algorithm was programmed for numerical calculations and tested for a number of objects
Интегралы и интегральные преобразования, связанные с векторным гауссовским распределением
This paper is dedicated to the integrals and integral transformations related to the probability density function of the vector Gaussian distribution and arising in probability applications. Herein, we present three integrals that permit to calculate the moments of the multivariate Gaussian distribution. Moreover, the total probability formula and Bayes formula for the vector Gaussian distribution are given. The obtained results are proven. The deduction of the integrals is performed on the basis of the Gauss elimination method. The total probability formula and Bayes formula are obtained on the basis of the proven integrals. These integrals and integral transformations could be used, for example, in the statistical decision theory, particularly, in the dual control theory, and as table integrals in various areas of research. On the basis of the obtained results, Bayesian estimations of the coefficients of the multiple regression function are calculated.Рассматриваются интегралы и интегральные преобразования, относящиеся к функции плотности вероятности векторного гауссовского распределения и возникающие в вероятностных приложениях. Представлены три интеграла, позволяющие рассчитывать моменты векторного гауссовского распределения, а также формулы полной вероятности и союз Байеса. Приводятся доказательства полученных результатов. Вывод интегралов выполнен на основе метода исключения Гаусса. Формулы полной вероятности и Байеса получены на основе доказанных интегралов. Представленные интегралы и интегральные преобразования могут быть использованы в различных вероятностных приложениях, например в теории статистических решений, в частности, в теории дуального управления, а также как табличные интегралы в различных областях исследований. На основе полученных результатов рассчитаны байесовские оценки коэффициентов множественной функции регрессии
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