The problem of finding linear unbiased estimates of the linear operator of unknown matrices — components of the observations vector, is investigated. It is assumed that the observation vector additively depends on a random vector with zero expected value, and the unknown correlation matrix belongs to a known bounded set. For the introduced class of linear estimates, necessary and sufficient conditions for the existence of solutions of operator equations that determine the unknown parameters of the vector estimate, are proved. The form of the guaranteed mean square error of the estimate is introduced on the sets of constraints of the problem parameters. The influence on the linear unbiased estimate of small perturbations of known rectangular matrices, which are the composites of the observations vector components, is also investigated. The analytical form is given through the parameters of the perturbed set of singularities for the introduced special operators that depend on a small parameter, which determine the corresponding operator equations, as well as their approximate solutions, in the first approximation of the small parameter method. A test example of solving the problem of finding a linear unbiased estimate under the condition of perturbation of both linearly independent and linearly dependent known observation matrices is presented.
Анотація. Розглянуто моделі поширення довільної кількості видів інформації, які подано у вигляді систем нелінійних диференціальних рівнянь зі стаціонар-ними параметрами. Проаналізовано різні способи подання спостережень. На-ведено алгоритми отримання усередненої оптимальної середньоквадратичної прогнозної оцінки та гарантованої прогнозної оцінки. Наведено приклад зна-ходження усередненої оптимальної середньоквадратичної оцінки для випадку поширення одного виду інформації та алгоритм знаходження гарантованих прогнозних оцінок для окремого випадку подання множини можливих похи-бок спостережень. Подано результати числового експерименту для задачі по-шуку гарантованих прогнозних оцінок для систем з двома джерелами інфор-мації.Ключові слова: моделі поширення інформації, прогнозні оцінки, невизначе-ність, середньоквадратичні оцінки. ВСТУПРозглядається деяка соціальна група чисельністю L , на яку спрямовується інформаційна дія (атака, вплив) по N каналах, причому кількість суб'єктів, що сприйняли інформацію k -го типу залежить як від зовнішньої дії, так і від спілкування суб'єктів між собою. Позначимо через ) (t x k кількість суб'єктів, що сприйняли інформацію k -го типу в момент часу t ; через ) (t k
We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average values based on observations of realizations of random matrix sequences. It is shown that such guaranteed estimates are obtained either as solutions to boundary value problems for systems of linear differential equations or as solutions to the corresponding Cauchy problems. We establish the form and look for errors for the guaranteed RMS quasi-minimax estimates of the special forecast vector and parameters of unknown average values. In the presence of small perturbations of known matrices in the model of matrix observations, quasi-minimax RMS estimates are found, and their guaranteed RMS errors are obtained in the first approximation of the small parameter method. Two test examples for calculating the guaranteed root mean square estimates and their errors are given.
This paper presents research of system of nonlinear equations. The algorithms of building a priory estimations, optimal functional estimations and guaranteed estimations of non-stationary parameters of differential equations are offered. These results are spread for discrete-time models. The algorithms of building optimal estimations and guaranteed estimations of non-stationary parameters of difference non-linear equations are offered. The approaches to construct optimal estimations based on Bellman functions and Kalman-Bussi filter. For each algorithms of building optimal functional estimations and guaranteed estimations the error of estimation are offered. We have presented as an example the results of numerical experiments to build guaranteed and optimal estimates for mathematical model of spreading one type of information with external influence. The number of information is taken as key parameter promoting accomplishment of aim. Information is spread in the community along internal (interpersonal communication of the members of social community) and external threads (mass media). Also for simplicity models with a constant number of individuals who are intentionally able to perceive and further spread an informational massage are explored. The model takes the form of non-linear ordinary differential equation with stationary parameters. The peculiarity of such models is that they allow a reasonable level of precision to model the subject area and obtain he results that can be uses in practice. The numerical experiments demonstrated the practical meaning of offered results. The offered approaches except theoretical interest has an important practical meaning. The results can be useful for algorithm development for estimation of dynamic of process in the information-communicative space.
The problems of guaranteed mean square estimation of unknown rectangular matrices based on observations of linear functions from random matrices with random errors are considered in the paper. Asymptotic distributions of guaranteed errors and guaranteed estimates are obtained in the case of small perturbations of the matrices. A test example of the asymptotic distribution is given.
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