Study on estimable parametric functions of balanced fractional factorial designs

在这项研究中，我们得到了非常密切的关系之间的矩阵方程组的解的存在条件和设计的某些决议来自一个简单的（部分）平衡阵列的存在条件。进一步给出了信息矩阵w.r.t.的不可约表示。一些代数的理想可以用Gram矩阵的乘积来描述，而Gram矩阵的元素仅由数组的后缀给出. 1.设计的构造和分析：（1）2^m-BFF设计情形：利用Gram矩阵的秩条件，得到了分辨率为R^*（{0，1}l3）和R^*（{1} l3）的2^m-BFF设计，其中，组件（或治疗组合）的数量（例如=N）小于不可忽略的因子效应的数量（例如=v_3）。利用某些结合方案的代数结构，给出了分辨率为R^*（{0，1}l3）的2 ^m-BFF设计的方差分析。(2)2^<m1+m2>-PBFF设计情况：在这种情况下，我们得到了所有分辨率为IV的2^<m1+m2>-B F设计，其中N<v且2[小于或等于] m_k[小于或等于] 4.2.最优性准则：作为一般A-和D-最优性准则的推广，我们分别提出了三种GA_α-和GD_α-最优性准则（α=0，1，2），反映了析因效应之间的混杂（或混叠）关系不可忽略，但对估计不感兴趣。3.最优设计：（1）2 μ m-BFF设计情形：得到了分辨率为R^*（{0，1}）13）的GAα和GDα最优2 μ m-BFF设计，其中6 ≤ m ≤ 8，N<v_3，此外，还得到了分辨率为R^*（{ 1}）~（13）的GAα-最优2 μ m-BFF设计，其中6 ≤ m ≤ 8，N<v_3。(2)2^<m1+m2>-PBFF设计情形：得到了分辨率为IV的GAα-最优2^<m1+m2>-PBFF设计，其中N<vand 2 [小于或等于]m_κ[小于或等于]4.

In this study, we obtained very closed relationship between the existence conditions of a solution in the system of the matrix equations and the existence conditions of a design of some resolution derived from a simple(partially)balanced array. Further-more the irreducible representation of the information matrix w.r.t. the ideals of some algebra can be described by the product of Gram matrix whose elements are given by only the suffix of an array.1.Construction and analysis of a designs : (1)2^m-BFF designs case : Using the rank conditions of the Gram matrix, we obtained 2^m-BFF designs of resolutions R^*({0,1}l3)and R^*({1})l3), where the number of assemblies(or treatment combinations)(=N, say)is less than the number of non-negligible factorial effects(=v_3, say). Furthermore using the algebraic structure of some association scheme, we presented the ANOVAof a 2^m- BFF design of resolution R^*({0,1}l3). (2) 2^<m1+m2>-PBFF designs case : In this case, we obtained all 2^<m1+m2>-B F F designs of resolution IV, where N<vand 2【less than or equal】 m_k【less than or equal】4.2.Optimality criteria : As a generalization of the ordinary A- and D-optimality criteria, we proposed three kinds of GA_α- and GD_α-0ptimality criteria each(α=0, 1,2), which reflect the confounded(or aliased)relationship among the factorial effects to be non-negligible but not to be interest in estimation. However these new optimality criteia will be reprodued by new methods, but now it is not be sure.3.Optimal designs : (1)2^m-BFF designs case : GAα and GDα optimal 2^m-BFF designs of resolution R^*({0,1})l3) ware obtained, where 6【less than or equal】m【less than or equal】8 and N<v_3, and in addition GAα-optimal 2^m-BFF designs of resolution R^*({ 1})l3)ware also obtained, where 6【less than or equal】m【less than or equal】8 and N<v_3. (2)2^<m1+m2>-PBFF designs case : We obtained GAα-potimal 2^<m1+m2>-PBFF designs of resolution IV, where N<vand2【less than or equal】m_κ【less than or equal】4.