多水平定量因子试验的设计与分析

It is essential to determine the criterion for comparing different designs before a proper design is chosen. When experiments with qualitative factors or two-level quantitative factors are considered, minimum aberration criterion is often recommended. However, for an experiment with high-level quantitative factors, it is not enough to choose a design only by minimum aberration. The current project will consider two approaches, i.e., beta-wordlength pattern under a polynomial model and uniform minimum aberration criterion. Inner requirements and combinatorial structures for best designs under beta-wordlength pattern and uniform minimum aberration designs will be discussed. By utilizing combinatorial design theory, coding theory, finite field property and stochastic optimization algorithm, effective construction methods for best design under beta-word length pattern and high-level low-discrepancy designs will be provided. A linear relationship between average uniformity and generalized word length pattern will be considered for arbitrary level, in order to extend the concept of uniform minimum aberration design. The project also aims to establish a relationship between uniform minimum aberration design and best design under beta-word length pattern, at least for some special types of regular designs, thus can verify the use of uniform minimum aberration designs and ability of uniformity for comparing efficiency and robust property of designs.

确定比较不同试验设计好坏的标准对于选择合适的设计来说是至关重要的。当所考虑的试验仅包含定性因子或二水平定量因子时，最小低阶混杂设计是值得推荐的。但对于多水平定量因子试验来说，仅仅采用最小低阶混杂准则来选择设计是不够的。本项目将分别讨论两种改进方案，即采用多项式模型下的beta-字长型为准则和采用均匀最小低阶混杂为准则来进行试验设计。项目将讨论beta-字长型最优设计与均匀最小低阶混杂设计的内在结构，并通过组合设计理论、编码理论、有限域性质及全局优化算法等方法给出更为有效的构造最优beta-字长型设计及多水平低偏差设计；对任意水平试验，建立平均均匀性度量与广义字长型之间的线性关系，进而拓展均匀最小低阶混杂设计概念；并对某些特殊类型的正则设计，建立均匀最小低阶混杂设计与最优beta-字长型设计之间的关系，在理论上说明均匀最小低阶混杂设计的合理性及均匀性度量在衡量设计效率和稳健性上的内在能力。

确定比较不同试验设计好坏的标准对于选择合适的设计来说是至关重要的。当所考虑的试验仅包含定性因子或二水平定量因子时，最小低阶混杂设计是值得推荐的。但对于多水平定量因子试验来说，仅仅采用最小低阶混杂准则来选择设计是不够的。.本项目主要研究了多水平定量因子试验的设计的性质分析和相关构造方法。项目针对三水平和四水平正则设计，分别给出beta-字长型的若干性质刻画及相应的最优设计构造方法。利用随机贪心算法, 通过编程实现对于给定参数下搜索在低阶混杂前提下的beta-字长型准则下最优设计的算法。对低阶混杂设计进行水平置换后最小中心化(可卷型)L2-偏差设计的搜索；另外也研究了其他特殊类型试验设计理论与相关的构造方法。本项目所研究的目标在执行过程中得以基本完成。目前已完成6篇SCI论文，包括Annals of Statistics, Biometrika, Journal of Statistical Planning and Inference, Journal of Complexity, Journal of Statistical Computation and Simulation , Acta Mathematica Sinica各一篇。有一篇文章被数学学报中文版接受，另外两篇已经完成写作，准备投到相应的SCI期刊。在项目执行过程中，结合研究内容，培养了9名学术型硕士研究生。期间，项目组负责人及成员积极参加国际国内相关学术会议并作报告，同时与国内外相关学者保持沟通和合作，并邀请诸多学者来苏州大学访问交流。