q分量二阶混料中心多项式模型V-最优设计

V-optimal designs for mixture central polynomial model of q-component second-degree

在混料试验设计与最优设计的研究中,对于常用最优准则的研究,理论已趋于成熟,且一般模型的D-最优设计和A-最优设计都已经研究得比较透彻。而对V-最优设计的研究相对较少,它的意义是回归方程预测方差在整个试验区域上平均值达到最小,主要的难点在于对整个试验区域上的积分。文章讨论了q分量二阶混料中心多项式模型的V-最优设计,为了得到该最优设计所对应的测度,主要运用分块矩阵的乘法运算、逆运算以及回归方程预测方差的积分运算。根据V最优准则,给出了相对应的条件极小值问题的目标函数,使用Lagrange乘子法并结合软件mathematica获得了该问题V-最优观测频数的一般解析表达式。最后利用软件mathematica得到了q=3时该二阶单纯形中心设计具体的V最优配置。

In the study of mixture experimental design and optimal design, for the research on common optimal criteria, the theory has become mature. And, the D-optimal design and A-optimal design of the general model have been studied thoroughly. However, there are relatively few V-optimal studies. Its significance is that the average value of the regression equation prediction variance reaches the minimum of the whole experimental area. Its main difficulty lies in the integration of the entire test area. This paper discusses the V-optimal design of the q-component second-order mixture center polynomial model. In order to obtain the measure corresponding to the optimal design, the multiplication operation, the inverse operation of the block matrix and the integral operation of the regression equation prediction variance are mainly used. According to the V-optimal criterion, the objective function of the corresponding conditional minima problem is given, and the general analytical expression of the V-optimal observation frequency of the problem is obtained by using the Lagrange multiplier method and the software mathematica. Finally, the software mathematica is used to obtain the specific V optimal configuration of the second-order simplex-center design when q=3.