The total method of Chebyshev interpolation in the problem of constructing a linear regression

A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The minimax criterion is used to estimate the measure of correction of the initial data; therefore, the proposed approach can be called the total method of Chebyshev approximation (interpolation). It leads to a nonlinear mathematical programming problem, which is reduced to solving a finite number of linear programming problems. This number depends exponentially on the number of parameters, therefore, some methods are proposed to overcome this problem. The results obtained are illustrated with practical examples based on real data, namely, the birth rate in the Federal Districts of the Russian Federation is analyzed depending on factors such as urban population, income and investment. Linear regression dependencies for two and three features are constructed. Based on the empirical fact of statistical stability (conservation of signs of the coefficients), the possibility of reducing the enumeration of linear programming problems is demonstrated.