Study on construction and analysis of balanced fractional3^m factorial designs-main effects being estimable

在本研究中，我们获得了与（部分）平衡分数（(P) BFF）设计相似的结果，这两个水平的研究得到了JSPS科学研究补助金(C)的部分支持，合同编号为14580348。在二能级情况下，信息矩阵的不可约表示可以用格拉姆矩阵的积来描述。与简单数组（SA）的索引有关的Gram矩阵的线性无关的列向量在2层SA中仅为1，而在3层SA中为1或2。设计的构造：(1)平衡二阶（3^m-BSO）设计：在二阶模型下，如果主效应的线性分量是可估计的，则设计的分辨率为R（{10}∪S|Ω）。具有足够项至少有一个0的SA称为“受限SA （RSA）”。一个分辨率为R（{10}∪S|Ω）的3^-BSO设计由一个除一个设计外N<v(m)的RSA构造，其中N为实验单元的总数，v(m)为不可忽略的阶乘效应的数目。(2)分辨率IV的3^m- bff设计：使用与(1)相似的方法，我们可以从N<v(m)的SA中获得分辨率IV的3^m- bff设计。在BFF设计课中，一般有14种分辨率IV设计。最优性准则：ga最优准则是在前面的研究中引入的。两因素相互作用（某些情况下包括一般平均值）之间的混淆（或混叠）关系用复数公式表示。为此，我们引入了新的最优性准则，称为“GA^*-最优性准则”。它是对ga -最优性准则的改进。优化设计：(1)3”-BSO设计：从N<v(m) w.r.t.的SA中导出的最优3^m-BSO设计获得了4〓m〓6的GA”-最优准则。(2) 3^m- bff设计：在这种情况下，SA的N<v(m)，而GA*-最优性准则尚未得到。少

In this study, we obtain similar results to the (partially) balanced fractional ((P) BFF) designs for the 2 levels whose research was partially supported by Grand-in-Aid for Scientific Research (C) of the JSPS under contract number 14580348. In the 2 levels case, the irreducible representation of the information matrix w.r.t. the ideals of some algebra can be described by the product of the Gram matrix. The linearly independent column vectors of the Gram matrix concerned with an index of a simple array (SA) are only one for the 2 levels SA, but for 3 levels SA, one or two.1. Construction of designs : (1) Balanced second-order (3^m-BSO) designs : Under the second-order model, if the linear component of the main effect are estimable, then a design is said to be of resolutions R({10}∪S|Ω). An SA with indices whose suffices have at least one zero is called a "restricted SA (RSA)". A 3^-BSO design of resolutions R({10}∪S|Ω) is constructed by an RSA with N<v(m) except for only one design, wh … More ere N is the total number of the experimental units and v(m) is the number of non-negligible factorial effects.(2) 3^m-BFF designs of resolution IV: Using the similar method mentioned (1), we can obtain 3^m-BFF designs of resolution IV derived from an SA with N<v(m). In the BFF design class, there are generally fourteen kinds of resolution IV designs.2. Optimality criterion: The GA-optimal criterion was introduced by the previous research mentioned above. The confounding (or aliasing) relations among the 2-factor interactions (some case include the general mean) are expressed as plural formulas. Thus we introduce new optimality criterion, which is called "GA^*-optimality criterion". It is an improvement of the GA-optimality criterion.3. Optimal designs : (1) 3"-BSO designs : Optimal 3^m-BSO designs derived from an SA with N<v(m) w.r.t. the GA"-optimality criterion are obtained for 4〓m〓6.(2) 3^m-BFF designs : In this case, optimal designs derived from an SA with SA with N<v(m) w.r.t. the GA*-optimality criterion are not obtained yet. Less