Normal System in Laplace Expansion and Related Regression Modeling Problems

This paper considers some innovative theoretical features and practical applications of the normal system of equations used for estimating parameters in multiple linear regression. The Laplace expansion of a determinant by cofactors and double Laplace expansion are employed for resolving the normal system. Additional features are described, including the ridge regularization applied directly to the normal system, geometric interpretation as a unique hyperplane through the points of special weighted means, Mahalanobis distances from observations to these means for the linear link functions, and multidimensional interpolation. The found properties are useful for a better understanding and interpretation of multiple regression, and the numerical examples demonstrate convenience and applicability of these tools in data modeling.