Geometric designs and rotatable designs, I
Geometric designs and rotatable designs, I
复制标题
几何设计和可旋转设计，我
DOI：
10.1007/s00373-021-02274-0
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发表时间：
2022
影响因子：
0.7
通讯作者：
K. Ito
中科院分区：
文献类型：
--
作者：
M. Sawa;M. Hirao;K. Ito
Victoir（2004）提出了一种构造具有少量点的高维立方体公式的强大方法，该方法包括针对高八面体群的角向量方法和基于正交阵列和组合设计的细化方法。Hirao et al.（2014）将此应用于三维单位球$${\varvec{B}}^d$$上的最优可旋转设计的构建，从而找到了一些具有少量点的无限级数的三阶设计。随后，Sawa和Hirao（2017）讨论了群的角向量方法，这些群可以实现为半超立方体的对称群。本文对这些工作作了进一步的推广和改进，并给出了少量点的d -最优可旋转设计的许多实例和无穷级数。该报告的核心和新颖之处在于对数值分析中的立方体理论或组合学中的欧几里得设计理论的关注，不仅改进和/或概括了这些数学领域的先前工作，而且在立方体理论的现代框架中对各种类型的析因设计（包括Box-Hunter固体设计、Box-Behnken设计、中心复合设计和Plackett-Burman设计）提供了统一的数学描述。例如，我们的新框架使我们能够简单地计算出这样的经典阶乘设计可进行常阶旋转的最大整数。
Victoir (2004) developes a powerful method for constructing high-dimensional cubature formulas with small number of points, which consists of the corner-vector approach for the hyperoctahedral groupand the thinning approach based on orthogonal arrays and combinatorial designs. Hirao et al. (2014) applies this to constructions of optimal rotatable designs on thed-dimensional unit ball $${\varvec{B}}^d$$ and thereby finds some infinite series of third-order designs with small number of points. Afterwards, Sawa and Hirao (2017) discusses the corner-vector approach for groupthat can be realized as the symmetry group of a demihypercube in. In the present paper we discuss a further generalization and improvement of these works, and describe many examples and infinite series ofD-optimal rotatable designs with small number of points. The core and novelty in the presentation lies in the focus on the cubature theory in numerical analysis or Euclidean design theory in combinatorics, which not only improves and/or generalizes previous works in those fields of mathematics but also provides a unified mathematical description of various classes of factorial designs, including Box–Hunter solid designs, Box–Behnken designs, central composite designs and Plackett–Burman designs, in the modern framework of the cubature theory. Our new framework enables one, for example, to briefly evaluate the maximum integertfor which such classical factorial designs are oft-th order rotatable.