Geodesic
Compromise Pareto's evaluation of parameters linear regression
Matematičeskoe modelirovanie, Tome 32 (2020) no. 11, pp. 70-78.

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

The material in this article is based on the author’s works on the construction of the Pareto set in the two-criterion problem of estimating the parameters of a linear regression equation with loss functions corresponding to the city distance and Chebyshev distance. It is known that the first of them is insensitive to emissions, while the second, on the contrary, gravitates to them. In these works, it was shown that such a problem reduces to a multicriteria linear programming problem, and its solution is the Pareto set. The article proposes a method for verifying the Paretiness of an arbitrary parameter estimation, as well as, in the case of identifying its non-parity, determining compromises in the given meanings of the estimates. Moreover, all the formulated problems are reduced to computationally simple linear programming problems.
Keywords: linear regression, two-criteria parameter estimation, urban distance, Chebyshev distance, Pareto set of estimates, Pareto compromise estimates. 
@article{MM_2020_32_11_a5,
     author = {S. I. Noskov},
     title = {Compromise {Pareto's} evaluation of parameters linear regression},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {70--78},
     publisher = {mathdoc},
     volume = {32},
     number = {11},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2020_32_11_a5/}
}
TY  - JOUR
AU  - S. I. Noskov
TI  - Compromise Pareto's evaluation of parameters linear regression
JO  - Matematičeskoe modelirovanie
PY  - 2020
SP  - 70
EP  - 78
VL  - 32
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2020_32_11_a5/
LA  - ru
ID  - MM_2020_32_11_a5
ER  - 
%0 Journal Article
%A S. I. Noskov
%T Compromise Pareto's evaluation of parameters linear regression
%J Matematičeskoe modelirovanie
%D 2020
%P 70-78
%V 32
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2020_32_11_a5/
%G ru
%F MM_2020_32_11_a5
S. I. Noskov. Compromise Pareto's evaluation of parameters linear regression. Matematičeskoe modelirovanie, Tome 32 (2020) no. 11, pp. 70-78. http://geodesic.mathdoc.fr/item/MM_2020_32_11_a5/

[1] Norman R., Draper, Harry Smith. Applied Regression Analysis, 3rd Ed., John Wiley Sons, Inc., 1998, 736 pp. | MR

[2] E. Forster, B. Ronz, Methods of Correlation and Regression Analysis, Finansy i statistika Publ., M., 1983, 304 pp.

[3] S. G. Radchenko, Metodologiya regressionnogo analiza, Kornijchuk, K., 2011, 376 pp.

[4] S. A. Ajvazyan, Prikladnaya statistika i osnovy ekonometriki, Yuniti, M., 2001, 432 pp.

[5] Z. Brandt, Analiz dannyh, Mir, M., 2003, 213 pp.

[6] https://soft.mydiv.net/win/download-gretl.html

[7] E. Z. Demidenko, Linejnaya i nelinejnaya regressii, Finansy i statistika, M., 1981, 304 pp. | MR

[8] S. I. Noskov, Tekhnologiya modelirovaniya ob'ektov s nestabil'nym funkcionirovaniem i neopredelennost'yu v dannyh, Oblinformpechat', Irkutsk, 1996, 320 pp.

[9] S. I. Noskov, Metodi modelirovaniya obektov s nestabilnim funkcionirovaniem i neopredelennostyu v dannish, dis. dokt. teh. nauk, NGU, Novosibirsk, 1994

[10] S. I. Noskov, “L-mnozhestvo v mnogokriterial'noj zadache ocenivaniya parametrov regressionnyh uravnenij”, Informacionnye tekhnologii i problemy matematicheskogo modelirovaniya slozhnyh sistem, 2004, no. 1, 64–71

[11] S. I. Noskov, A. V. Baenhaeva, “Mnozhestvennoe ocenivanie parametrov linejnogo regressionnogo uravneniya”, Sovrem. tekhnologii. Sist. analiz. Modelir., 2016, no. 3, 133–138

[12] G. K. Kamenev, “Multicriteria Identification Sets Method”, Computational Mathematics and Mathematical Physics, 56:11 (2016), 1843–1858 | DOI | MR | Zbl

[13] G. K. Kamenev, “A Multicriteria Method for Identification and Forecasting”, Mathematical Models and Computer Simulations, 10:2 (2018), 154–163 | DOI | MR | Zbl

[14] A. V. Lakeev, S. I. Noskov, “Metod naimen'shih modulej dlya linejnoj regressii. Chislo nulevyh oshibok approksimacii”, Sovrem. tekhnologii. Sistemn. analiz. Modelir., 2012, no. 2, 48–50

[15] L. Yu, M. Zeleny, “The set of all nondoinated solutions in linear cases and multicriteria simplex method”, J. of Math. Anal. and Applic., 49:2 (1975), 430–468 | DOI | MR | Zbl

[16] A. V. Lotov, V. A. Bushenkov, G. K. Kamenev, O. L. Chernih, Kompyuter i poisk kompromissa. Metod dostijimih celei, Nauka, M., 1997, 239 pp. | MR

[17] A. V. Lotov, V. A. Bushenkov, G. K. Kamenev, Interactive Decision Maps. Approximation and Visualization of Pareto Frontier, Appl. Optimization, 89, Kluwer Acad. Publ., Boston–Dordrecht–New York–London, 2004, 310 pp. | DOI | MR | Zbl

[18] P. Harold Benson, “An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem”, Journal of Global Optimization, 13 (1998), 1–24 | DOI | MR | Zbl

[19] Andreas Lohne, Vector Optimization with Infimum and Supremum, Springer Science Business Media, 2011, 206 pp. | MR | Zbl

[20] A. V. Lotov, I. I. Pospelova, Konspekt lekcii po teorii i metodam mnogokriterialnoi optimizacii, M., 2005

[21] A. V. Baenhaeva, M. P. Bazilevskij, S. I. Noskov, “Programmnyj kompleks mnozhestvennogo ocenivaniya regressionnyh modelej”, Informacionnye tekhnologii i problemy matematicheskogo modelirovaniya slozhnyh sistem, 2016, no. 17, 38–44

[22] S. A. Ajvazyan, I. S. Enyukov, L. D. Meshalkin, Prikladnaya statistika. Osnovy modelirovaniya i pervichnaya obrabotka dannyh, Finansy i statistika, M., 1983, 472 pp. | MR