Pseudo inverse matrix research, as well as related issue of their application, is in focus of scientific literature recently. Moreover, the field of applications of pseudo inverse matrices is expanding and obtained theoretical results are successfully used to solve practical problems. Interest in the pseudo inverse matric problem is large due to their various applications: in control theory, identification problems, approximation theory, statistical problems, theory of recurrent filtration and others. We investigate a linear operator equation, where the operator is a linear combination of matrices. We obtain solution using the pseudo inverse operation and small parameter method. Research subject of this publication is the application of the pseudo inverse matrix technique for solving linear regression analysis problems under the conditions when matrices are observing elements. Such problems appear in recognition and classification tasks with matrix features. The first variant of an application of pseudo-inversion matrix in matrix Euclidean spaces is Klein formula [1, 2]. We note that among the publications of recent years there is a work [7] devoted to an application of pseudo inverse technique in matrix Euclidean spaces.
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.
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.
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