SJR: 0.275 SNIP: 0.59 CiteScore™: 0.8

ISSN 打印: 1064-2315
ISSN 在线: 2163-9337

# 自动化与信息科学期刊

DOI: 10.1615/JAutomatInfScien.v52.i1.10
pages 1-12

## Perturbation Method in Problems of Linear Matrix Regression

Alexander G. Nakonechnyi
Kiev National Taras Shevchenko University, Kiev
Grigoriy I. Kudin
Kiev National Taras Shevchenko University, Ukraine
Petr N. Zinko
Kiev National Taras Shevchenko University, Kiev
Taras P. Zinko
Kiev National Taras Shevchenko University, Kiev

### ABSTRACT

Within the frameworks of linear regression this work studied estimates linear by observations, in particular the unbiased ones that leads to unbiased equations, among solutions of which the ones minimum in norm are distinguished, that allows us to minimize the root mean square error at uncorrelated observation errors with equal variances. Preliminary the problem of linear regression analysis is presented in the form of linear operator in a space of independent rectangular matrices connected with the equation of linear functions unbiasedness from matrix parameters. It is assumed that for this operator in the unperturbed version its SVD representation is known as well as SVD representation of the pseudo inverse to it operator. Since it is necessary to determine the singular set of the perturbed operator for determining eigenvalues and eigenvectors of the special symmetric matrix we employ the perturbation method, according to the general theory of operators in Euclidean space we determine the eigenmatrices of conjugate perturbed operator. Assuming that the problem of the linear regression analysis at the presence of perturbation of observation matrices can be solved in real time conditions, the resulting formulas are given in the first approximation of the small parameter. We present the test example which besides a small parameter includes parameters of random perturbations.

1. Albert A., Regression, pseudo inversion and recurrent valuation [Russian translation], Nauka, Moscow, 1977. .

2. Draper N., Smith H., Applied regression analysis [Russian translation], Izdatelskiy dom "Williams", 2007. .

3. Kirichenko N.F., Donchenko V.S., Pseudoinversion in clustering problems, Kibernetika i sistemnyi analiz, 2007, No. 4, 73-92. .

4. Kirichenko N.F., Kudin G.I., Analysis and synthesis of systems of signals classification by means of perturbations of pseudoinverse and projection operations, Kibernetika i sistemnyi analiz, 2009, No. 3, 47-57. .

5. Donchenko V.S., Zinko T., Skotarenko F., "Feature vectors" in grouping information problem in applied mathematics: vectors and matrices, Problems of computer intellectualization, ITHEA, Kyiv, 2012, 111-124. .

6. Nayfeh A.H., Introduction to perturbation technique [Russian translation], Mir, Moscow, 1984. .

7. Blokhintsev D.I., Foundations of quantum mechanics [in Russian], Nauka, Moscow, 1976. .

8. Nakonechniy O.G., Kudin G.I., Zinko T.P., Formulas of perturbation for one class of pseudoinverse operators, Matematychni studii, 2019, 52, No. 2, 124-132. .

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