Orthogonal polynomial multidimensional-matrix regression analysis

Abstract

The article is devoted to the orthogonal regression analysis, which is associated with the representation of the regression function by Fourier series by the multidimensional-matrix (mdm) orthogonal polynomials, in opposite to the (usual) regression analysis, when the regression function is approximated by the (usual) polynomial (by the degrees of the independent mdm input variable). We will also distinguish the classical regression analysis, when the scalar or might the classical vector-matrix mathematical approaches are used, and the mdm regression analysis, when the mdm variables and the mdm mathematical approach are used. In this article, the orthogonal regression analysis is developed on the base of the orthogonal polynomials and the mdm mathematical approach, so called the mdm orthogonal polynomial regression analysis. The known results from the theory of the orthogonal mdm polynomials and Fourier series of the vector argument are generalized to the case of the mdm argument and function. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for the potential studies are obtained. The general case of the approximation of the mdm function of the mdm argument by the Fourier series is realized programmatically as the single program function and its efficiency is confirmed by the computer calculations. The properties of the estimations of regression coefficients and unknown parameters are studied and their distributions when the normal distribution of the measurement’s errors are obtained for the arbitrary covariance matrix of the errors of measurements and the arbitrary degree of the approximating polynomial. These results allow testing the hypothesis and building the hyper-rectangular confidence areas relating the orthogonal regression function. Theoretical results are confirmed by computer simulation.