GEOMETRIC MEANING OF LEAST SQUARES METHOD

Abstract

The geometric interpretation of the least squares method is presents. At the same time, one of its possible generalizations to multidimensional space is proposed. This approach makes it possible to expand the capabilities one of the key methods of multidimensional approximation and effectively use it for geometric modeling of multifactor processes and phenomena. The analytical description of the proposed method is performed using point equations. The geometric interpretation of the generalized least squares method, which consists in determining the linear surface of the minimum width between two hypersurfaces in the hyperspace of the General position, extends the tools of geometric modeling objects in multidimensional space and can be effectively used for geometric modeling of multifactor processes and phenomena’s, by presenting them in the form of geometric multiparameter objects passing through predetermined points. In this case, the approximation process is reduced to determining the coordinates of the nodal points of the geometric object of multidimensional space that satisfy the condition of minimizing the sum of the lengths of the segments between the nodal points and the given ones. Also, using the proposed approach, the generalization of the coefficient of determination on the multidimensional space as a tool for assessing the accuracy of the results of multidimensional approximation is performed. An example of using the proposed generalization for the geometric modeling of a three-parameter hypersurface of the response belonging to a four-dimensional space in relation to the determination of strength characteristics over the entire volume of a concrete column is given.