Maximizing the number of acceptable errors in linear regression approximation with deterministic constraints

Abstract

The paper considers the problem of constructing a linear regression model aimed at maximizing the number of observations for which the approximation error does not exceed a given threshold value. Unlike classical methods that minimize the sum of squares or error modules, the proposed approach allows you to directly control the proportion of acceptable deviations. The problem is formulated in the form of a linear-Boolean programming (LBP) model. In addition to the basic model, six types of deterministic constraints are proposed that reflect a priori expert knowledge about the subject area: interval constraints on parameters, constraints on the sum and maximum of absolute errors, sign compatibility of coefficients, the requirement of accurate hits (the presence of a given number of observations with zero error) and the uniformity of the contribution of features across observations. It is shown that all the introduced restrictions allow linearization, which preserves the belonging of the problem to the LBP class. The statement about the equivalence of the initial and linearized statements is formulated and proved. The proposed approach is used to model the passenger turnover of the Russian Federation's air transport based on statistical data for 2002-2019. A comparative analysis with previously published results shows that the introduction of additional restrictions leads to a tightening of the allowable set of solutions: the number of allowable errors decreases from 11 to 8, however, the sum of error modules decreases from 175.6 to 167.8, which indicates an increase in the accuracy of approximation over the entire set of observations.