Обчислювальні методи для оцінювання параметрів у багатофакторній регресії

Abstract

Регресійний аналіз є основним статистичним методом побудови математичних моделей об’єктів або явищ за експериментальними даними. Визначення невідомих параметрів регресії зводиться до розв’язування систем лінійних рівнянь. У зв'язку з різноманітністю й специфікою обчислюваних матриць, свої особливості мають і відповідні методи розв’язування систем лінійних алгебраїчних рівнянь (СЛАР). У роботі запропоновано чисельні методи для розв’язування однієї з важливих задач регресійного аналізу. Проведено оцінювання складності методу та аналіз похибок заокруглення.

Regression anlysis is the powerful method of modern statistics and can be applied in various areas. It is a basic tool for development of mathematical model basing on experimental information. Many techniques for carrying out regression analysis have been developed. Available methods such as linear regression and ordinary least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data. The performance of regression analysis methods in practice depends on the form of the data generating process, and how it is related with the regression approach being used. Since the true form of the data-generating process is generally not known, regression analysis often depends to some extent on making assumptions about this process to some extent. These assumptions are sometimes testable if a sufficient quantity of data is available. Determining of unknown coefficients in multiple linear regression is provided by solving of simultaneous equations and therefore it is convenient to use matrix models for it. Taking into account variety and specifics of calculated matrices, appropriate methods of simultaneous equations solving possess their own unique features. Programs developed for such tasks are not typical in standard computer software and are specially made for a specific problem. There are several important problems which occur during development of numeric methods for simultaneous equations, called calculation errors. They are caused by rounding which takes place at any arythmetic operation on a computer, inaccuracy of input information about system, and related problems of memory saving and reducing of necessary number of operations. There are many effective methods of solving simultaneous equations with numerical elements. Such methods require different approaches and algorithms. This paper proposes numerical methods for solving one of the important problems of regression analysis. Estimation of the method complexity and approximation errors have also been executed.