Modelling and Data Analysis

2023. Vol. 13, no. 3, 52–65

doi:10.17759/mda.2023130304

ISSN: 2219-3758 / 2311-9454 (online)

# Identification of Possible Estimates Areas for Parameters of Fully connected Linear Regression Models

## Abstract

This article is devoted to the study of fully connected linear regression models, in which the observed variables contain errors, and the pairs of true variables are interconnected by linear functional dependencies. When estimating fully connected regressions, the main problem is the correct choice of the error variances ratios of the variables. If the choice is made incorrectly, then the fully connected regression estimates will be biased. The purpose of this article is to find the dependence of main parameters possible estimates areas on the possible error variances ratios of the variables in fully connected regressions. For the first time, with the help of matrix algebra elements, the inverse problem is solved - analytical dependences of the error variances ratios of variables on the main parameters are obtained. These dependences make it possible to identify the parameters possible estimates areas in which the necessary condition for the extremum of the objective function is satisfied. It is proved that, under certain conditions, for any error variances ratios of the variables, the parameters estimates always lie inside an open convex polygon located only in one of the orthants of the multidimensional space. In this case, the signs of the estimates always agree with the signs of the corresponding correlation coefficients. A numerical experiment was carried out, confirming the correctness of the results obtained.

## General Information

**Keywords:** errors-in-variables model, fully connected linear regression model, weighted total least squares, parameter estimation, convex polygon

**Journal rubric:** Optimization Methods

**Article type:** scientific article

**DOI:** https://doi.org/10.17759/mda.2023130304

**Received:** 10.07.2023

**Accepted:**

**For citation:**
Bazilevskiy M.P. Identification of Possible Estimates Areas for Parameters of Fully connected Linear Regression Models. *Modelirovanie i analiz dannikh = Modelling and Data Analysis,* 2023. Vol. 13, no. 3, pp. 52–65. DOI: 10.17759/mda.2023130304. (In Russ., аbstr. in Engl.)

## References

- Montgomery D.C., Peck E.A., Vining G.G. Introduction to linear regression analysis. John Wiley & Sons, 2021.
- Xu P. Improving the weighted least squares estimation of parameters in errors-in-variables models, Journal of the Franklin Institute, 2019, vol. 356, no. 15, pp. 8785–8802. DOI:10.1016/j.jfranklin.2019.06.016
- Demidenko E.Z. Lineynaya i nelineynaya regressii [Linear and nonlinear regressions]. Moscow, Finansy i statistika, 1981. 304 p.
- Golub G.H., Van Loan C.F. An analysis of the total least squares problem, SIAM Journal on Numerical Analysis, 1980, vol. 17, no. 6, pp. 883–893.
- Bazilevskiy M.P. Metody postroeniya regressionnykh modeley s oshibkami vo vsekh peremennykh [Methods for constructing errors-in-variables regression models]. Irkutsk, IrGUPS, 2019. 208 p.
- Deming W.E. Statistical adjustment of data. New York, Wiley, 2011. 288 p.
- Koh N.W.X., Markus C., Loh T.P., Lim C.Y. Comparison of six regression-based lot-to-lot verification approaches, Clinical Chemistry and Laboratory Medicine, 2022, vol. 60, no. 8, pp. 1175–1185. DOI:10.1515/cclm-2022-0274
- Bazilevskiy M.P. Issledovanie povedeniya otnositel'nykh vkladov peremennykh v obshchuyu determinatsiyu v otsenennom na osnove metoda vypryamleniya iskazhennykh koeffitsientov regressionnom uravnenii [Researching the behavior of variables relative contributions to the total determination in regression equation estimated using the method of distorted coefficients straightening], The Herald of the Siberian State University of Telecommunications and Information Science, 2022, no. 1(57), pp. 89–96.
- Bazilevskiy M.P. Mnogofaktornye modeli polnosvyaznoy lineynoy regressii bez ogranicheniy na sootnosheniya dispersiy oshibok peremennykh [Multifactor fully connected linear regression models without constraints to the ratios of variables errors variances], Informatics and Applications, 2020, vol. 14, no. 2, pp. 92–97. DOI:10.14357/19922264200213
- Bazilevskiy M.P. Metod vypryamleniya iskazhennykh iz-za mul'tikollinearnosti koeffitsientov v regressionnykh modelyakh [Method of straightening distorted due to multicollinearity coefficients in regression models], Informatics and Applications, 2021, vol. 15, no. 2, pp. 60–65. DOI:10.14357/19922264210209

## Information About the Authors

## Metrics

##### Views

Total: 59

Previous month: 3

Current month: 0

##### Downloads

Total: 61

Previous month: 6

Current month: 1