Generalization of Non-elementary Linear Regressions

Earlier, the author developed a non-elementary linear regression consisting of a linear part and all possible combinations of min and max binary operations. This article is devoted to its generalization. For the first time a non-elementary linear regression with a linear part and all possible combinations of binary, ternary, ..., l-ary operations min and max has been introduced. The proposed model generalizes both linear regression and the Leontief function, and can be effectively used both for predicting and for interpreting the study object functioning. An estimation algorithm was developed using the method of least squares for non-elementary linear regressions without a linear part and with an l-ary operation min (max), i.e. regressions with specification in the form of a Leontief function. The essence of the algorithm is to form a set of possible values of slope coefficients, from which a point is selected with the minimum value of the residual sum of squares. A system of linear inequalities is identified that makes it possible to form such a set. Using the algorithm, a model of the gross regional product of the Irkutsk region was construct and its interpretation was given.

1. Khenrik B., Dzhozef R., Mark F. Mashinnoe obuchenie [Machine Learning]. Saint Petersburg, Piter, 2017. 336 p.
2. Flakh P. Mashinnoe obuchenie. Nauka i iskusstvo postroeniya algoritmov, kotorye izvlekayut znaniya iz dannykh [Machine Learning. The Art and Science of Algorithms that Make Sense of Data]. Moscow, DMK Press, 2015. 400 p.
3. Molnar C.Interpretable machine learning. Lulu. com, 2020.
4. Doshi-Velez F., Kim B. Towards a rigorous science of interpretable machine learning. arXiv preprint arXiv:1702.08608, 2017.
5. Montgomery D. C., Peck E. A., Vining G. G. Introduction to linear regression analysis. John Wiley & Sons, 2021.
6. Keith T. Z. Multiple regression and beyond: An introduction to multiple regression and structural equation modeling. Routledge, 2019.
7. Gelman A., Hill J., Vehtari A. Regression and other stories. Cambridge University Press, 2020.
8. Brachunova U. V. Chislennoe modelirovanie zaryadnogo balansa legkovogo avtomobilya [Numerical simulation of the charging balance of a passenger car], Proceedings of the TSU. Technical Sciences, 2022, no. 9, pp. 453–458.
9. Yarymbash D. S., Kotsur M. I., Yarymbash S. T., Kilimnik I. M. Modelirovanie elektromagnitnykh protsessov pri rabote silovykh transformatorov pod nagruzkoy i v rezhime kholostogo khoda [Electromagnetic Processes Simulation of Power Transformers in Operation and in No-load Mods], Problemele Energeticii Regionale, 2020, no. 1 (45), pp. 1–13.
10. Balgarina L., Dzhumabaev S., Shokamanov Yu. Proizvodstvennaya funktsiya Kobba–Duglasa: opyt primeneniya v Severo-Kazakhstanskoy oblasti [Cobb – Douglas Production Function: application experience in the North Kazakhstan region], Economic Series of the Bulletin of the L.N. Gumilyov ENU, 2022, vol. 141, no. 4.
11. Chesnokov E. A. Sravnenie regressionnykh modeley ekonomicheskogo razvitiya Rossii [Comparison of regression models of economic development in Russia], Moscow economic journal, 2021, no. 7, pp. 96–105.
12. Bazilevskiy M.P. Postroenie stepenno-pokazatel'nykh i lineyno-logarifmicheskikh regressionnykh modeley [Constructing power-exponential and linear-logarithmic regression models], Control Sciences, 2021, no. 3, pp. 25–32.
13. Reva S. A., Arnautov A. V., Klitsenko O. A., Petrov S. B. Prognosticheskaya znachimost' logisticheskoy regressionnoy modeli dlya otsenki riska retsidiva u bol'nykh rakom predstatel'noy zhelezy posle radikal'noy prostatektomii [Prognostic significance of the logistic regression model for assessing the risk of recurrence in patients with prostate cancer after radical prostatectomy], Research'n Practical Medicine Journal, 2022, vol. 9, no. 4, pp. 96–105.
14. Kokoulina M. V., Epifanova A., Pelinovskiy E. N., Kurkina O. E., Kurkin A. A. Analiz dinamiki rasprostraneniya koronavirusa s pomoshch'yu obobshchennoy logisticheskoy modeli [Analysis of coronavirus dynamics using the generalized logistic model], Proceedings of NSTU n.a. R.E. Alekseev, 2020, no. 3 (130), pp. 28–41.
15. Kleyner G. B. Proizvodstvennye funktsii: Teoriya, metody, primenenie [Production functions: Theory, methods, application]. Moscow: Finance and Statistics, 1986. 239 p.
16. Bazilevskiy M. P. Otsenivanie lineyno-neelementarnykh regressionnykh modeley s pomoshch'yu metoda naimen'shikh kvadratov [Estimation linear non-elementary regression models using ordinary least squares], Modeling, optimization and information technology, 2020, vol. 8, no. 4 (31).
17. Bazilevskiy M. P. Otbor informativnykh operatsiy pri postroenii lineyno-neelementarnykh regressionnykh modeley [Selection of informative operations in the construction of linear non-elementary regression models], International Journal of Open Information Technologies, 2021, vol. 9, no. 5, pp. 30–35.
18. Bazilevskiy M. P. Metod postroeniya neelementarnykh lineynykh regressiy na osnove apparata matematicheskogo programmirovaniya [A method for constructing nonelementary linear regressions based on mathematical programming], Control Sciences, 2022, no. 4, pp. 3–14.
19. Noskov S. I., Khonyakov A. A. Programmnyy kompleks postroeniya nekotorykh tipov kusochno-lineynykh regressiy [Software complex for building some types pieces of linear regressions], Information technology and mathematical modeling in the management of complex systems, 2019, no. 3 (4), pp. 47–55.
20. Bazilevskiy M. P. Otsenka metodom naimen'shikh kvadratov prosteyshikh neelementarnykh lineynykh regressiy s lineynym argumentom v binarnoy operatsii [Ordinary least squares estimation of simple non-elementary linear regressions with a linear argument in a binary operation], Proceedings in Cybernetics, 2022, no. 4 (48), pp. 69–76.
21. Noskov S. I. Tekhnologiya modelirovaniya ob"ektov s nestabil'nym funktsionirovaniem i neopredelennost'yu v dannykh [Technology for modeling objects with unstable operation and uncertainty in data]. Irkutsk, RITs GP «Oblinformpechat'», 1996. 320 p.