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

O. G. Nakonechnyi; P. M. Zinko

doi:10.30970/ms.60.2.208-222

O. G. Nakonechnyi Taras Shevchenko National University of Kyiv Faculty of Computer Science and Cybernetics Kyiv, Ukraine
P. M. Zinko Taras Shevchenko National University of Kyiv Faculty of Computer Science and Cybernetics Kyiv, Ukraine

DOI: https://doi.org/10.30970/ms.60.2.208-222

Keywords: observation;, linear matrix estimator;, linear operator;, conjugate operator;, random linear operator;, operator equation;, correlation matrix;, scalar product of matrices;, guaranteed rms matrix estimate;, guaranteed rms error;, minimax estimate;, quasiminimax estimate.

Abstract

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.

Author Biographies

O. G. Nakonechnyi, Taras Shevchenko National University of Kyiv Faculty of Computer Science and Cybernetics Kyiv, Ukraine

Taras Shevchenko National University of Kyiv
Faculty of Computer Science and Cybernetics
Kyiv, Ukraine

P. M. Zinko, Taras Shevchenko National University of Kyiv Faculty of Computer Science and Cybernetics Kyiv, Ukraine

Taras Shevchenko National University of Kyiv
Faculty of Computer Science and Cybernetics
Kyiv, Ukraine

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