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

在这项研究中，我们得到了类似于两个水平的(部分)平衡分数((P)BFF)设计的结果，该设计的研究部分得到了合同号为14580348的JSPS科学研究大援助(C)的支持。在两个水平的情况下，信息矩阵的不可约表示W.r.t.某些代数的理想可以用Gram矩阵的乘积来描述。与简单数组(SA)的索引相关的Gram矩阵的线性独立列向量对于2级SA只有一个，但对于3级SA是一个或两个。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可估，则称设计具有R({10}∪S|Ω)解。一个指数至少为零的SA称为“受限SA(RSA)”。3^-BSO设计的分辨率R({10}∪S|Ω)是由一个具有N&lt；v(M)的RSA构造的，除了一个设计，wh…这里N是实验单元的总数，v(M)是不可忽略的析因效应的个数。(2)解IV的3^m-bff设计：用上述(1)类似的方法，我们可以得到由N&v(M)的SA导出的解IV的3^m-bff设计。在最好的设计类中，通常有14种分辨率IV设计。最优性准则：GA-最优准则是在前人研究的基础上提出的。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)关系被表示为复数公式。为此，我们引入了一种新的最优性准则，称为“GA^*-最优性准则”。这是对遗传算法最优性准则的改进。最优设计：(1)3“-BSO设计：由具有N&lt；v(M)w.r.t的SA导出的最优3^m-BSO设计。对于4〓m〓6，得到GA”-最优准则。(2)3^m-bff设计：在这种情况下，从具有N&lt；v(M)w.r.t的SA得到的最优设计。目前还没有得到GA*-最优性准则。较少

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