编码映射与折叠反转在试验设计中的应用研究

Design of experiment is widely used in many fields as an important branch of mathematical statistics. Fractional factorial designs are very popular in view of cost-effectiveness, however, one consequence of using a fractional factorial design is the aliasing among factorial effects. Foldover is universally accepted and widely used as an effective technique to dealias. The latest studies show that the constructed designs have excellent statistical properties by the mapping between binary and quanternary code in code theory. On the basis of the existing results, the project aims at studying the optimal foldover and related problems under uniformity criterion by using the mapping between binary and quanternary code. Firstly, the relationships between a four-level design and the corresponding two-level design under the uniform criterion and aberration criterion will be studied thoroughly via the mapping between binary and quanternary code. Secondly, the foldover strategy of four-level designs will be built and the optimal foldover plans of four-level designs will be studied based on the mapping between binary and quanternary code.Thirdly, the above results will be extended to mixed two- and four-level designs. Finally, the construction of the optimal designs will be considered via the method of foldover and code mapping.The research results of the project will enrich the theory of fractional factorial designs and uniform designs, and provide convenience and guide for the practical application.

试验设计作为数理统计的重要分支，在许多领域应用十分广泛。考虑到试验花费的有效性，部分因析设计受到极大地关注，但使用部分因析设计会产生因子效应别名。折叠反转作为解除别名因子效应的有效方法之一，已获得公认和广泛应用。最新研究表明利用编码理论中的二、四元编码间的映射所构造的设计具有很好的统计性质。本项目将在已有的研究基础上，旨在利用二、四元编码间的映射研究在均匀性准则下的最优折叠反转及相关问题：（一）基于二、四元编码间的映射，深入研究四水平设计和相应的两水平设计之间在均匀性准则、混杂准则下的关系；（二）基于二、四元编码间的映射，构建四水平设计的折叠反转策略并研究其最优折叠反转方案；（三）把前述研究结果推广到二、四混水平设计中；（四）将折叠反转、编码映射的思想和技术运用于优良设计的构造。本项目的研究结果将进一步丰富和完善部分因析设计、均匀设计的理论，为实际应用提供便利和指导。

试验设计作为数理统计的重要分支，在许多领域应用十分广泛。考虑到试验花费的有效性，部分因析设计受到极大地关注，但使用部分因析设计会产生因子效应别名。折叠反转作为解除别名因子效应的有效方法之一，已获得公认和广泛应用。最新研究表明利用编码理论中的二 、四元编码间的映射所构造的设计具有很好的统计性质。本项目在已有的研究基础上，旨在利用二、四元编码间的映射研究在均匀性准则下的最优折叠反转及相关问题。本项目主要开展了以下三方面的研究：(1) 基于编码映射构造高水平对称、非对称设计。(2)基于折叠反转构造扩大设计。(3)结合编码映射和折叠反转构造优良设计。通过该项目的研究，获得很多理论结果和优良设计表，这些研究成果完善、丰富了因子设计的理论，获得的优良设计表为实际应用提供便利喝指导。本项目共完成学术论文19篇，其中在SCI检索期刊发表论文13篇；CSCD收录6篇；指导硕士研究生3名；获湖南省自然科学奖三等奖1项；主持召开了一次学术会议。