Multiple Regression Model as Interpolation Through the Points of Weighted Means

Abstract

A well-known property of the multiple linear regression is that its plane goes through the point of the mean values of all variables, and this feature can be used to find the model's intercept. This work shows that a regression by n predictors also passes via n additional points of the specific weighted mean values. Thus, the regression is uniquely defined by all these n+1 multidimensional points of means, and approximation of observations by the theoretical model collapses to the interpolation function going through the knots of the weighted means. This property is obtained from the normal system of equations which serves for finding the linear regression parameters in the ordinary least squares approach. The derived features can be applied in nonlinear modeling for adjusting the model parameters so that the fitted values would go through the same reference points of means, that can be useful in applied regression analysis. Numerical examples are discussed. The found properties reveal the essence of regression function as hyperplane going through special points of mean values, which makes regression models more transparent and useful for solving and interpretation in various applied statistical problems.