Алгоритмы построения дискретных A-оптимальных планов эксперимента при активной идентификации регрессионных моделей многофакторных систем

Abstract

To solve the problem of effective identification of regression models of multifactor systems, as a rule, they resort to using the concept of optimal experiment design. The synthesis of experimental designs involves the use of an a priori chosen optimality criterion. Quite a few criteria have been proposed. Most often, criteria are used that are associated with the accuracy of estimating the parameters of regression models. We can name such well-known criteria as: the D-optimality criterion, the A-optimality, criterion and the E-optimality criterion. It should be noted that most of the theoretical and applied research is associated with the use of the D-optimality criterion. It is noted in the paper that often plans built according to the A-optimality criterion show good performance for a number of other optimality criteria. At the same time, the criterion itself characterizes an average variance of estimates of the parameters of the regression model and, for A-optimal designs the dispersion ellipsoid has the smallest overall dimensions. The use of the D-optimality criterion makes it possible to obtain an ellipsoid of dispersion of parameter estimates of the smallest volume, which does not exclude the possibility of obtaining an ellipsoid elongated along one or more principal axes. The paper proposes and describes two algorithms for the synthesis of discrete A-optimal designs. The first of them is based on the concept of the consistent completion of the experiment design to the required volume developed by the author. It can be successfully used in a situation where the researcher needs to increase the number of experiments to achieve the required accuracy of the resulting model. The second algorithm, which makes it possible to build plans for a given number of observations, consists of iterations in which points are added and removed from the plan according to certain rules.