多因素系统回归模型主动辨识中离散a -最优实验设计的构建算法

A. A. Popov
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引用次数: 0

摘要

为了解决多因素系统回归模型的有效识别问题,他们通常采用最优实验设计的概念。实验设计的综合涉及使用先验选择的最优性准则。已经提出了相当多的标准。大多数情况下,使用的标准与估计回归模型参数的准确性有关。我们可以将这些众所周知的标准命名为:d -最优性标准、a -最优性标准和e -最优性标准。应该指出的是,大多数理论和应用研究都与d -最优性准则的使用有关。本文指出,通常根据a -最优性准则构建的规划对于许多其他最优性准则都表现出良好的性能。同时,该准则本身的特征是回归模型参数估计的平均方差,对于a -最优设计,色散椭球具有最小的总体尺寸。利用d -最优性准则可以得到最小体积参数估计的色散椭球,这并不排除得到沿一个或多个主轴延长的椭球的可能性。本文提出并描述了两种离散a -最优设计的综合算法。其中第一种是基于实验设计一致完成的概念,由作者开发的所需体积。它可以成功地用于研究人员需要增加实验次数以达到所需精度的结果模型的情况。第二种算法可以为给定数量的观测建立计划,它由迭代组成,在迭代中,根据一定的规则从计划中添加和删除点。
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Algorithms for constructing discrete A-optimal experiment designs in active identification of regression models of multifactor systems
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.
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