Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
KIBERNETYKA TA SYSTEMNYI ANALIZ
International Theoretical Science Journal
-->

DOI 10.34229/KCA2522-9664.24.6.10
UDC 519.711
O.G. Nakonechnyi1, P.M. Zinko2, T.P. Zinko3


1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

oleksandrnakonechny@knu.ua

2 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

petro.zinko@knu.ua

3 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

taras.zinko@knu.ua

ESTIMATES OF THE SOLUTIONS OF OPERATOR EQUATIONS UNDER UNCERTAINTY

Abstract. The paper focuses on finding the guaranteed estimates for rectangular matrices that satisfy the linear operator equations with unknown right-hand sides. Under certain assumptions, it is shown that guaranteed estimates of matrices can be found as the solutions of systems of operator equations. In the case where observation data are given with deterministic errors, which belong to certain bounded sets, we obtain the expressions for upper and lower estimates of the solutions of such equations, as well as expressions for upper and lower errors of such estimates. The obtained results are illustrated by an example.

Keywords: linear operator equation, operator core, conjugate operator, guaranteed a posteriori estimate of matrix, guaranteed a posteriori error of the estimate of matrix, minimum by norm solution of the equation.


full text

REFERENCES

  • 1. Yue S. Application of random matrix theory in statistics and machine learning. Publicly Accessible Penn Dissertations, 2021. 240 p. URL: https://repository.upenn.edu/edissertations .

  • 2. Ke Y., Minsker S., Ren Z., Sun Q., Zhou W.-X. Uzer-friendly covariance estimation for heavy-tailed distributions. Statistical Science. 2019. Vol. 34 (3). P. 454–471. URL: https://doi.org/ 10.48550/arXiv.1811.01520 .

  • 3. Lopes R.C. Robust estimation of the mean a random matrix: A non-asymptotic study. Centro de Investigacion en Matematicas, A.C., 2020. 187 p.

  • 4. Battey H., Fan J., Lu J., Zhu Z. Distributed testing and estimation under sparse high dimensional models. The Annals of Statistics. 2018. Vol. 46(3). P. 1352–1382. URL: .

  • 5. Cai T.T., Wei H. Distributed Gaussian mean estimation under communication constraints: Optimal rates and communication-efficient algorithms. arXiv: 2001.08877v1, 2020. URL: https://doi.org/10.1214/17-AOS1587 .

  • 6. Ke T., Ma Y., Lin X. Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis. arXiv: 2006.00436, 2020. URL: https://doi.org/10.48550 .

  • 7. McKennan C. Factor analysis in high dimensional biological data with dependent observations. arXiv: 2009.11134, 2020. URL: https://doi.org/10.4855 .

  • 8. Chatterjee S. Matrix estimation by universal singular value thresholding. Annals of Statistics. 2015. Vol. 43, N 1. P. 177–214. URL: https://doi.org/10.48550 .

  • 9. Kapustian O.A., Nakonechnyi O.G. Approximate minimax estimation of functionals of solutions to the wave equation under nonlinear observations. Cybernetics and Systems Analysis. 2020. Vol. 56, N 5. P. 793–801. URL: https://doi.org/10.1007 .

  • 10. Nakonechniy A.G., Kudin G.I., Zin'ko P.N., Zin'ko T.P. Minimax mean square estimates of matrix parameters in linear regression problems under uncertainty. Problemy upravleniya i informatiki. 2021. N 4. P. 28–37

  • 11. Nakonechniy O.H., Kudin G.I., Zinko P.M., Zinko T.P. Guaranteed rms estimates of observations with unknown matrices. Zhurnal obchyslyuvalʹnoyi ta prykladnoyi matematyky. 2022. N 2. P. 98–115

  • 12. Nakonechniy O.G., Kudin G.I., Zinko P.M., Zinko T.P., Shusharin Y.V. Guaranteed root mean square estimates of linear matrix equations solutions under conditions of uncertainty. Mathematical Modeling and Computing. 2023. Vol. 10, N 2. P. 474–486. URL: https://doi.org/10. 23939 .

  • 13. Nakonechniy O.G., Kudin G.I., Zinko P.M., Zinko T.P. Guaranteed root mean square estimates of the forecast of matrix observations under conditions of statistical uncertainty. System Research and Information Technologies. 2023. N 2. P. 86–103. URL: https://doi.org/10.20. 535 .

  • 14. Nakonechniy O.G., Zinko P.M. Estimates of matrix solutions of operator equations with random parameters under of uncertainties. Mathematychni Studii. 2023. Vol. 60, N 2. P. 208–222. URL: https://doi.org/10.30970 .




© 2024 Kibernetika.org. All rights reserved.