Perturbation Method in Problems of Linear Matrix Regression

Nakonechnyi, Oleksandr; Kudin, G. I.; Zinko, Petr N.; Zinko, Taras

doi:10.1615/jautomatinfscien.v52.i1.10

Cited by 4 publications

(4 citation statements)

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“…and from inequality (14) we obtain the formula for calculating the lower bound of the ratio of the squared minimax and quasi-minimax estimation errors…”

Section: Now We Can Write Down the Equality Minmentioning

confidence: 99%

“…In our previous papers [14][15][16][17], the problem of linear estimation in the space of rectangular observation matrices was solved when the known matrices are perturbed. The operator equations for the coefficients of the vector linear estimate and the dependences of the linear estimates on small perturbations of the matrix coefficients of the linear regression were obtained.…”

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confidence: 99%

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Estimates of matrix solutions of operator equations with random parameters under uncertainties

Nakonechnyi,

Zinko

2023

Mat. Stud.

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We investigate problems of estimating solutions of linear operator equations with random parameters under conditions of uncertainty. We establish that the guaranteed rms estimates of the matrices are found as solutions of special optimization problems under certain observations of the system state. As the output signals of the system, we have observations that are described by linear functions from the solutions of such equations with random right-hand sides, which have unknown second moments. Under the condition that the observation second moments of the right-hand parts and errors belong to certain sets, it is proved that the guaranteed estimates are expressed through solutions of operator equation systems. When the linear operator is given by the scalar product of rectangular matrices, a quasi-minimax estimate and its error are constructed. It is shown that the quasi-minimax estimation error tends to zero when the number of observations tends to infinity. An example of calculating the guaranteed rms estimate of the matrix's trace, which is a solution of a matrix equation with a random parameter, is given.

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“…and from inequality (14) we obtain the formula for calculating the lower bound of the ratio of the squared minimax and quasi-minimax estimation errors…”

Section: Now We Can Write Down the Equality Minmentioning

confidence: 99%

mentioning

confidence: 99%

Estimates of matrix solutions of operator equations with random parameters under uncertainties

Nakonechnyi,

Zinko

2023

Mat. Stud.

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“…This article examines estimates of unknown mathematical expectations based on observations of realizations of random matrix sequences. Scientific publications [1][2][3][4][5][6][7][8][9][10][11][12][13][14], in which estimates of distribution parameters were studied, are devoted to the problems of matrix sequence statistics. We formulate and solve new problems of estimating the mean values of random matrix sequences.…”

Section: Introductionmentioning

confidence: 99%

Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty

Nakonechnyi¹,

Kudin²,

Zinko³

et al. 2023

SRIT

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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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“…A number of matrix evaluation problems under conditions of statistical uncertainty were investigated in the authors' works [11][12][13]. Also, in publications [14][15][16], the problems of estimating matrices with a small parameter are solved.…”

Section: Introductionmentioning

confidence: 99%

Guaranteed root mean square estimates of linear matrix equations solutions under conditions of uncertainty

Nakonechnyi¹,

Kudin²,

Zinko³

et al. 2023

Math. Model. Comput.

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The article focuses on the linear estimation problems of unknown rectangular matrices, which are solutions of linear matrix equations with the right-hand sides belonging to bounded sets. The random errors of the observation vector have zero mathematical expectation, and the correlation matrix is unknown and belongs to one of two bounded sets. Explicit expressions of the guaranteed root mean square errors of estimates for linear operators acting from the space of rectangular matrices into some vector space are given. Guaranteed quasi-minimax root mean square errors of linear estimates are obtained. As the test examples, two options for solving the problem are considered, taking into account small perturbations of known observation matrices.

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Perturbation Method in Problems of Linear Matrix Regression

Cited by 4 publications

References 0 publications

Estimates of matrix solutions of operator equations with random parameters under uncertainties

Estimates of matrix solutions of operator equations with random parameters under uncertainties

Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty

Guaranteed root mean square estimates of linear matrix equations solutions under conditions of uncertainty

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