区组设计的最小低阶投影均匀性理论及应用研究

In the areas of modern industrial and agricultural production, biological medicine, aerospace, scientific experiment and so on, we often face some complicated systems. Block designs and uniform designs are very effectively statistical experiment methods to deal with these complicated systems. Projective uniformity is one of important properties of design of experiments, and there are many fundamental problems deserve further study in depth. On the basis of the existing works, this project aims to systematically study the projective uniformity theory and statistical properties of block designs. Firstly, the theory and method of projective uniformity of two-level block designs will be investigated thoroughly. Secondly, for the complicated structure of multilevel and mixed-level designs, new measures of projective uniformity will be proposed based on the latest measures of uniformity such as generalized discrete discrepancy and so on. Thirdly, the theory of projective uniformity of multilevel and mixed-level block designs will be studied based on the new measures of projective uniformity. Fourthly, the relationships between projective uniformity criterion and the other existing optimal criteria of block designs will be studied. Furthermore, optimal block designs with minimum projection uniformity and mixed-level uniform designs will be constructed, and the applications and data analysis methods of the constructed designs will be investigated. The objective of this project is to build up the general theory of block designs with minimum projection uniformity, and to construct some designs and computer experiments possessed excellent properties. These research results will enrich and improve the theory of factorial designs and uniform designs, and provide convenience and guidance for the practical applications.

近代工农业生产、生物医药、航空航天、科学试验等领域往往面临一些复杂系统，区组设计、均匀设计是处理这些复杂系统非常有效的统计试验方法。投影均匀性是试验设计的重要性质，有许多基础问题需要深入研究。本项目将在已有的研究基础上，旨在对区组设计的投影均匀性理论和统计性质展开研究：（一）研究两水平区组设计的投影均匀性理论和方法；（二）针对多水平、混水平设计的结构复杂性，基于广义离散偏差等最新的均匀性测度建立新的投影均匀性测度；（三）在新的投影均匀性测度下探讨多水平、混水平区组设计的投影均匀性理论；（四）研究投影均匀性准则与区组设计中诸优良性准则之间的关系，最优投影均匀性的区组设计及混水平均匀设计的构造，并探讨所构造的设计的应用和数据分析方法。本项目的目标是建立区组设计最小低阶投影均匀性的一般理论，并构造一些优良设计和计算机试验，进一步丰富和完善区组设计、均匀设计的理论，为实际应用提供便利和指导。

近代工农业生产、生物医药、航空航天、科学试验等领域往往面临一些复杂系统，区组设计、均匀设计是处理这些复杂系统非常有效的统计试验方法。投影均匀性是试验设计的重要性质，有许多基础问题需要深入研究。本项目在已有的研究基础上，对区组设计、扩展设计的投影均匀性理论和统计性质展开研究：(1)两水平区组设计和组合设计的均匀性；(2)多水平和混水平设计的投影均匀性测度；(3)多水平、混水平扩展设计的均匀性；(4)高水平均匀设计及混水平均匀设计的构造方法及应用。本项目的研究建立了区组设计、扩展设计最小低阶投影均匀性准则，并构造一些优良设计和计算机试验，进一步丰富和完善区组设计、均匀设计的理论，为实际应用提供便利和指导。本项目共完成学术论文24篇，其中在SCI检索期刊发表论文13篇；指导完成硕士学位论文5篇；获湖南省自然科学奖三等奖1项；主持召开了两次学术会议；项目主持人入选湖南省121创新人才。