几种小型试验设计的构造与建模

Small experiments are widely used in engineering and many other related fields because lots of factors can be studied with few runs, as a result, one can get some useful results economically in a very short period of time. Most research is based on the assumption that all the response variables are normally distributed with the same variance now, there are many kinds of response coming from different background however. So this project aims to study the design and analysis of some small experimental designs under more realistic and practical conditions. In the project, we would consider the effect screening for supersaturated designs when the response variables are distributed discretely firstly by the theory of generalized linear model. Furthermore, a methodology would be developed to add runs to existing supersaturated designs and the technique should use information from the analysis of the initial experiment to choose the best possible follow-up runs. Based on the discussion, two-stage designs are involved incorporating other criteria to further reduce the dependency on the assumed model. At last, the optimality criteria, the constructions of designs and data analysis methods for supersaturated experiments with prior information will be discussed through the combination of Bayesian theory and background of the experiments. The contents of the project are proposed according to the demand of real-life cases and the most active topics in the current international studies, some design tables with good properties and open source code of data analysis approaches will be provided in order to directly benefit engineers and other practical users. Furthermore, the analysis of such experimental data which involves the theory of small sample may stimulate the proceeding of related statistical fields.

小型试验可用较低的成本在短时间内获得有用的分析结果，因此在工业生产中广为应用。目前对此类设计的研究多基于响应来自于同方差的正态分布这一假定，在一定程度上尚不能满足实际需要。本项目旨在更一般的符合工业生产实践的条件下，以提高质量、降低变差为优化目标，对于一些小型试验，研究其设计的构造与数据建模问题。内容包括：结合超饱和设计的特点，利用广义线性模型理论，对试验响应来自于离散型分布或者方差非齐性的情形，进行效应的筛选与建模研究；对于一般的超饱和设计，考虑最优跟随试验的设计，进而考虑两阶段的稳健超饱和设计；对于具备各类先验信息的试验，本项目将充分考虑试验的特点和背景，结合贝叶斯思想，研究最优设计的构造与数据建模问题。本项目的研究内容是结合实际问题需要及当前国际前沿动态提出来的，其成果将对实际工作者有一定的指导作用，涉及到的建模方法也会对统计中的小样本问题有一定的理论价值。

随着各种科技工程实际应用领域的快速发展，出现了越来越多的新问题, 对试验设计提出了越来越高的要求,快速获得数据进行分析对于研发会带来成本的节约。本项目对试验设计中的几种小型设计进行深入研究，考虑相关设计的最优性理论、构造及数据分析方法，取得了一定的科研成果。具体地，对于超饱和设计，提出一种基于贝叶斯的最优的追加试验方案，该准则同样适用于其他类型设计；对于Kriging模型，提出了一种变量选择方法，该方法的性能可由Gibbs sampler的收敛性来保证，且大量的随机模拟也显示我们的新方法与现有方法相比表现令人满意；对分片拉丁超立方体设计，提出了两种构造具有折叠反转结构的近似正交SLHDs的方法，该方法简便、高效，可容纳任意数量的片，在保证正交或近似正交的前提下，又具有较好的空间填充性质；对于多个分区组变量的区组设计，研究了2-水平最优设计理论；除了这些设计理论的研究，项目组还将统计理论和方法应用到实际领域，对我国部分地区的水文变化规律进行分析。在SCI检索期刊发表和在线发表论文6篇，协助指导完成硕士论文二篇，博士论文一篇。