Use of primary decomposition of polynomial ideals arising from indicator functions to enumerate orthogonal fractions
Use of primary decomposition of polynomial ideals arising from indicator functions to enumerate orthogonal fractions
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DOI：
10.1007/s42081-022-00149-z
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发表时间：
2021-09
影响因子：
1.3
通讯作者：
S. Aoki;M. Noro
中科院分区：
文献类型：
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作者：
S. Aoki;M. Noro
设计的多项式指示函数是由Fontana等人（J Stat Plan Inference 87:149 - 172,2000）首先引入的，用于两级情况。给出了指示函数的结构，特别是与设计正交性的关系。这些结果由Aoki （J Stat Plan Inference 203:91 - 105,2019）对一般多级情况进行了推广。作为这些结果的应用，我们可以用计算代数软件列举所有具有给定大小和正交性的正交分数因子设计。例如，Aoki（2019）给出了强度为3的设计的正交分数的分类，这是通过简单的变量消去得来的。然而，这种朴素方法的计算可行性取决于问题的大小。事实上，据报道，Aoki（2019）没有进行强度为2的设计的正交分数的计算。本文利用初等分解理论，对强度为2的设计的正交分数进行了枚举和分类。我们发现有35200个设计的正交半分数，强度为2，分为63个等效类。
A polynomial indicator function of designs is first introduced by Fontana et al. (J Stat Plan Inference 87:149–172, 2000) for two-level cases. They give the structure of the indicator functions, especially the relation to the orthogonality of designs. These results are generalized by Aoki (J Stat Plan Inference 203:91–105, 2019) for general multi-level cases. As an application of these results, we can enumerate all orthogonal fractional factorial designs with given size and orthogonality using computational algebraic software. For example, Aoki (2019) gives classifications of orthogonal fractions ofdesigns with strength 3, which is derived by simple eliminations of variables. However, the computational feasibility of this naive approach depends on the size of the problems. In fact, it is reported that the computation of orthogonal fractions ofdesigns with strength 2 fails to carry out in Aoki (2019). In this paper, using the theory of primary decomposition, we enumerate and classify orthogonal fractions ofdesigns with strength 2. We show there are 35,200 orthogonal half fractions ofdesigns with strength 2, classified into 63 equivalent classes.