# DIRECT SOLUTION OF POLYNOMIAL REGRESSION OF ORDER UP TO 3

2022;
: pp. 35-42
Автори:
1
Національний університет “Львівська політехніка”

This article presents results related to the direct solution of the polynomial regression parameters based on the analytical solving of regression equations. The analytical solution is based on the normalization of the values of independent quantity with equidistance steps. The proposed solution does not need to directly solve a system of polynomial regression equations. The direct expressions to calculate estimators of regression coefficients, their standard deviations, and also standard and expanded deviation of polynomial functions are given. For a given number of measurement points, the parameters of these expressions have the same values independently of the range of input quantity. The proposed solution is illustrated by a numerical example used from a literature source.

[1] M.G. Kendall and A. Stuart. (1966) The advanced theory of statistics. Vol. 2. Inference and relationship. Second Edition. Chares Griffin and Company Limited, London. https://archive.org/ details/in.ernet.dli.2015.212877.
[2] DRAPER, N. and SMITH, H. (1981) Applied Regression Analysis. 2nd ed., Wiley, New York. https://onlinelibrary.wiley.com/doi/book/10.1002/97 81118625590
[3] J.O. Rawlings, S. G. Pantula, D. A. Dickey. (1998) Applied Regression Analysis: A Research Tool, Second Edition, Springer-Verlag. New York, Berlin, Heidelberg. https://doi.org/10.1007/b98890.
https://doi.org/10.1007/b98890
[4] Handbook of Applicable Mathematics. Volume VI: Statistics, Part A, (1984). Edited by Emlyn Lloyd, John Wiley and Sons. https://www.amazon.com/ Handbook-Applicable-Mathematics-Statistics6/dp/0471900249.
[5] J.H. Pollard. (1977) Handbook of Numerical and Statistical Techniques. Cambridge University Press. https://cc.bingj.com/cache.aspx?q=J.H.+Pollard.+(1 977)+Handbook+of+Numerical+and+Statistical+Te chniques.&d=4579616625657925&mkt=enWW&setlang=en-US&w=lYKqOOVXlYsikZDqV7p9HJrRgt2oDA3.
[6] Applied Linear Regression, 3rd ed. Willey, Hoboken (2005). https://openlibrary.org/books/ OL3306403M/Applied_linear_regression.
[7] J. R. Taylor (1982). An introduction to error analysis. University Science Books Mill Valley, California 1982. https://openlibrary.org/books/ OL3786923M/An_introduction_to_error_analysis
[8] B. Forbes, (2009) Parameter estimation based on leastsquares methods. In F. Pavese and A. B. Forbes, editors, Data modeling for metrology and testing in measurement science, New York. Birkauser-Boston. https://link.springer.com/chapter/10.1007/978-0-8176- 4804-6_5.
[9] G. Mejer. (2008) Smart Sensor Systems. John Wiley, 2008. https://onlinelibrary.wiley.com/doi/book/ 10.1002/9780470866931/
https://doi.org/10.1002/9780470866931
[10] Evaluation of measurement data-Guide to the expression of uncertainty in measurement Joint Committee for Guides in Metrology, JCGM 100: 2008. https://www.iso.org/sites/JCGM/GUM/ JCGM100/.
[11] JCGM 101:2008. Evaluation of measurement data- Supplement 1 to the 'Guide to the Expression of Uncertainty in Measurement'- propagation of distributions using a Monte Carlo method'. BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML. https://www.bipm.org/documents/20126/2071204/J CGM.
[12] Dorozhovets M. Study of the effect of correlation of observation results on the uncertainty of linear regression. Pomiary, Automatyka, Kontrola. N12, 2008, s.31-34 (in Polish). https://bibliotekanauki.pl/ issues/8204.
[13] Dorozhovets M. Include measurement uncertainty of both quantities in linear regression. Pomiary, Automatyka, Kontrola. N9, 2008, pp.-612-615 (in Polish). https://bibliotekanauki.pl/issues/8089.