大型二四混水平均匀设计的构造及其性质

Construction and Properties of Mixed Two-and Four-Level Uniform Designs with Large Run Sizes

Doubling、tripling和quadrupling方法可用于构造大型的具有优良性质的二水平、三水平和四水平的空间填充设计.均匀设计作为一种空间填充设计被应用到许多领域.Li,et al.(2021)结合doubling和tripling方法构造大型二三混水平均匀设计.文章结合doubling和quadrupling方法将Li,et al.(2021)的结果推广到二四混水平情形,首先基于小规模二四混水平设计给出了二四混水平翻倍设计的结构;其次建立了翻倍设计的广义字长型和初始设计的广义字长型模式的解析关系;最后,以可卷型L2-偏差为均匀性度量讨论了翻倍设计和初始设计之间的均匀性关系,并分别获得翻倍设计和初始设计的可卷型L_(2)-偏差的两个下界.这些理论结果为构造具有优良性质的大型二四混水平均匀设计提供了理论基础.

Doubling,tripling and quadrupling can be used to construct large two-level,three-level and four-level space-filling designs with excellent properties.Uniform designs are widely applied in many fields as one of space-filling designs.Li,et al(2021)combined doubling with tripling to construct mixed two-and three-level uniform designs with large run sizes.In this paper,the results of Li,et al(2021)are extended to mixed two-and four-level case by combining doubling with quadrupling.Firstly,the construction of the mixed two-and four-level amplified design is presented based on the mixed two-and four-level design with small run sizes;Secondly,the analytical connection of generalized word-length pattern between the amplified design and the initial design is established;Finally,the uniformity between the amplified design and the initial design is discussed in terms of the wrap-around L_(2)-discrepancy,and two lower bounds of the amplified design and the initial design are obtained respectively.These theoretical results provide a theoretical basis for constructing nice mixed tw-oand four-level uniform designs with large run sizes.