Geodesic
The total method of Chebyshev interpolation in the problem of constructing a linear regression
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 52-63.

Voir la notice de l'article provenant de la source Math-Net.Ru

A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The minimax criterion is used to estimate the measure of correction of the initial data; therefore, the proposed approach can be called the total method of Chebyshev approximation (interpolation). It leads to a nonlinear mathematical programming problem, which is reduced to solving a finite number of linear programming problems. This number depends exponentially on the number of parameters, therefore, some methods are proposed to overcome this problem. The results obtained are illustrated with practical examples based on real data, namely, the birth rate in the Federal Districts of the Russian Federation is analyzed depending on factors such as urban population, income and investment. Linear regression dependencies for two and three features are constructed. Based on the empirical fact of statistical stability (conservation of signs of the coefficients), the possibility of reducing the enumeration of linear programming problems is demonstrated.
Keywords: data processing, linear regression, matrix correction, minimax criterion, linear programming problem. 
@article{CHEB_2022_23_4_a4,
     author = {V. A. Gorelik and T. V. Zolotova},
     title = {The total method of {Chebyshev} interpolation in the problem of constructing a linear regression},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {52--63},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a4/}
}
TY  - JOUR
AU  - V. A. Gorelik
AU  - T. V. Zolotova
TI  - The total method of Chebyshev interpolation in the problem of constructing a linear regression
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 52
EP  - 63
VL  - 23
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a4/
LA  - ru
ID  - CHEB_2022_23_4_a4
ER  - 
%0 Journal Article
%A V. A. Gorelik
%A T. V. Zolotova
%T The total method of Chebyshev interpolation in the problem of constructing a linear regression
%J Čebyševskij sbornik
%D 2022
%P 52-63
%V 23
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a4/
%G ru
%F CHEB_2022_23_4_a4
V. A. Gorelik; T. V. Zolotova. The total method of Chebyshev interpolation in the problem of constructing a linear regression. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 52-63. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a4/

[1] Eremin I. I., Theory of linear optimization. Inverse and ill-posed problems series, VSP, Utrecht–Boston–Koln–Tokyo, 2002 | MR

[2] Gorelik V. A., “Matrichnaya korrektsiya zadachi lineinogo programmirovaniya s nesovmestnoi sistemoi ogranichenii”, Zhurn. vychisl. matem. i matem. fizika, 41:11 (2001), 1697–1705 | MR | Zbl

[3] Gorelik V. A., Erokhin V. I., Optimalnaya matrichnaya korrektsiya nesovmestnykh sistem lineinykh algebraicheskikh uravnenii po minimumu evklidovoi normy, VTs RAN, M., 2004 | MR

[4] Gorelik V. A., Erokhin V. I. Pechenkin R. V., Chislennye metody korrektsii nesobstvennykh zadach lineinogo programmirovaniya i strukturnykh sistem uravnenii, VTs RAN, M., 2006

[5] Gorelik V. A., Trembacheva O. S., “Reshenie zadachi lineinoi regressii s ispolzovaniem metodov matrichnoi korrektsii v metrike $l_1$”, Zhurn. vychisl. matem. i matem. fizika, 56:2 (2016), 202–207 | DOI | Zbl

[6] Back A., “The matrix-restricted total least squares problem”, Signal Process, 87:10 (2007), 2303–2312 | DOI

[7] Hnětynková I., Plešinger M., Sima D. M., Starakoš Z., Van Huffel S., “The total least squares problem in $AX \approx B$: A new classification with the relationship to the classical works”, SIAM Journal on Matrix Analysis and Applications, 32:3 (2011), 748–770 | DOI | MR

[8] Hnětynková I., Plešinger M., Žáková J., “On TLS formulation and core reduction for data fitting with generalized models”, Linear Algebra and Its Applications, 577 (2019), 1–20 | DOI | MR

[9] Hnětynková I., Plešinger M., Žáková J., “Solvability classes for core problems in matrix total least squares minimization”, Applications of Mathematics, 64:2 (2019), 103–128 | DOI | MR

[10] Markovsky I., Van Huffel S., “Overview of total least-squares methods”, Signal Processing, 87:10 (2007), 2283–2302 | DOI | Zbl

[11] Meng L., Zheng B., Wei Y., “Condition numbers of the multidimensional total least squares problems having more than one solution”, Numerical Algorithms, 84:3 (2020), 887–908 | DOI | MR | Zbl

[12] Shklyar S., “Consistency of the total least squares estimator in the linear errors-in-variables regression”, Modern Stochastics: Theory and Applications, 5:3 (2018), 247–295 | DOI | MR | Zbl

[13] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 1, Fakty. Modeli, MTsNMO, M., 2016

[14] Gorelik V. A., Zolotova T. V., “Method of parametric correction in data transformation and approximation problems”, Lecture Notes in Computer Science (LNCS), 12422, 2020, 122–133 | DOI | MR

[15] Regiony Rossii. Sotsialno-ekonomicheskie pokazateli, 2019 (Data obrascheniya: 30 oktyabrya 2021) https://rosstat.gov.ru