Characterizing D-optimal Rotatable Designs with Finite Reflection Groups

被引:0
作者
Sawa M. [1 ]
Hirao M. [2 ]
机构
[1] Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, 657-8501, Kobe
[2] Department of Information and Science Technology, Aichi Prefectural University, 1522-3 Ibaragabasama, Nagakute, 480-1198, Aichi
来源
Sankhya A | 2017年 / 79卷 / 1期
基金
日本学术振兴会;
关键词
Approximate design; Optimal design; Euclidean design; Corner-vector method; D-optimality; Finite irreducible reflection group; Gaffke-Heiligers theorem; Invariant harmonic polynomial; 62K05; 65D32; 05E99;
D O I
10.1007/s13171-016-0091-1
中图分类号
学科分类号
摘要
We establish a powerful construction of D-optimal Euclidean designs, or D-optimal rotatable designs, on the unit hyperball by using the corner vectors associated with the symmetry groups of (semi-)regular polytopes. This is a full generalization of the classical construction choosing points in the form (a,…,a,0,…,0) or in their orbits under the symmetry group of a regular hyperoctahedron (Gaffke and Heiligers 1995b), (Hirao et al. 2014), as well as Scheffé’s {n, 2}-lattice design on the simplex. We prove a Gaffke-Heiligers type theorem for Dn- and An-invariant D-optimal Euclidean designs which is a “reduction theorem” on the computational cost of searching observation points, and thereby construct many families of D-optimal Euclidean designs. For each group An, Dn, Bn, H3, H4, F4, E6, E7, E8, we determine the maximum degree of a D-optimal Euclidean design constructed by our method and in particular discover examples of degrees 5 and 6 for E8 and H4, respectively. We also classify such maximum-degree designs for the groups H3, H4 and F4 acting on the 3- and 4-dimensional Euclidean spaces. © 2016, Indian Statistical Institute.
引用
收藏
页码:101 / 132
页数:31
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