ALGORITHMS FOR CONSTRUCTING A REGRESSION LINEAR WITH RESPECT TO UNKNOWN COEFFICIENTS ON A LIMITED AMOUNT OF EXPERIMENTAL DATA

Abstract

This publication continues the series of scientific works of the authors on the creation of algorithms for constructing multivariate regressions which are linear with respect to unknown coefficients by using linear programming models. To simplify the simulation modeling of their efficiency, we present the algorithms for the multivariate linear regression problem. The use of linear programming models requires minimizing the sum of the absolute differences used in the general procedure of the least squares method. The estimates of the unknown coefficients obtained as a result of solving the linear programming problem are linear with respect to the vector of the values ​​of the regression model in the statistical experiment. It is known that, by virtue of the Markov theorem, the estimates of the unknown coefficients obtained by the general procedure of the least squares method are efficient in the class of linear unbiased estimates. Thus, it would seem that the transition from the least squares method to the least absolute deviations used in the least squares method is a priori unproductive. But this is not so. From the proof of the Markov theorem, it follows that the linear estimation matrix must be constant and independent of the values of the regression model in the statistical experiment. The estimates obtained by the least absolute deviations method do not meet this condition. Indeed, the estimation matrix is the optimal basis for solving the linear programming problem by the simplex method and depends on the values of the regression model in the statistical experiment. Such a formulation of the problem allows introducing, into the optimization model, linear constraints that use the results of statistical tests and implement additional properties of the searched multivariate regression. The first studies of these algorithms have shown their efficiency, this allowed the authors to set the task of creating such algorithms that can not only compete with the general algorithmic procedure of the least squares method, but also be efficient for the case of a limited volume of experimental data, when the ratio of the average absolute value of the realizations of a random factor in the experiment to the average absolute value of the true regression on it is a sufficiently large value. In this case, it is incorrect to raise the problem of finding estimates of unknown coefficients that practically do not differ from the true ones, but, as experiments and, in particular, the examples given in this paper have shown, it is possible to find sufficiently good estimates of the average values of the true regression in the experiments conducted, which can be used, for example, in diagnosing the early stages of the onset of an epidemic of various diseases or in other recognition tasks.