Fractional Factorial Designs: Minimum Aberration and Related Topics

郑清水DMS-0071438摘要最小象差一直是选择好的部分析因设计的公认标准。这项研究研究了最小像差准则在几种不同设置下的变化，包括将实验单位分组到更均匀的块以提高精度的区块设计，以及在每个区块内某些因素保持不变的裂区设计。当一些因素需要比其他因素更大的实验单位时，或者当某些因素的影响不是主要感兴趣的时候，分裂图就会出现，但它们被包括在实验中，以研究它们与其他因素的相互作用。后者在质量改进的稳健参数设计中有非常重要的应用。这些设置有它们自己的特殊功能，需要不同的优化标准。已有的工作没有解决分裂图结构中存在两个不同错误的问题。这项研究的最终目的是在各种环境下获得关于最优设计结构的一般结果，并开发实用的算法来构建能够结合用户提供的先验知识和要求的设计。对于前者，首席调查者使用了编码论和有限射影几何的工具。最后，在新引入的广义最小像差准则下，研究了包括过饱和设计在内的非正则设计。实验设计在科学和工业研究中得到了广泛的应用。在工业实验中，往往要研究大量的因素，但进行实验的费用很高。在这种情况下，所谓的部分析因设计，其中只观察到所有可能的组合的一小部分，是特别有用的。近年来，析因设计受到了相当大的关注，这主要是因为它在工业制造中成功地应用于提高质量和生产率的试验。这项研究是为了研究如何构建有效的设计来提取更多的信息，特别是当需要消除系统的变异源(如实验材料的异质性或日常环境变化)以提高精度时。

Cheng, Ching-ShuiDMS-0071438AbstractMinimum aberration has been a well accepted criterion for choosing good fractional factorial designs. This research studies variants of the minimum aberration criterion in several different settings including block designs in which the experimental units are grouped into more homogeneous blocks to improve the precision, and split-plot designs in which some factors are held constant within each block. Split plots arise when some factors require larger experimental units than others, or when the effects of certain factors are not of major interest, but they are included in the experiment to study their interactions with other factors. The latter has a very important application to robust parameter designs in quality improvement. These settings have their own special features that call for different optimality criteria. Existing work did not address the issue that there are two different errors in split-plot structures. The ultimate goal of this research is to obtain general results on the structures of optimal designs in various settings, and to develop useful algorithms for constructing designs which can incorporate user-supplied prior knowledge and requirements. For the former, the Principal Investigator uses tools from coding theory and finite projective geometry. Finally, nonregular designs, including supersaturated designs, are studied under a newly introduced criterion of generalized minimum aberration.Experimental design is used extensively in a wide range of scientific and industrial investigations. In industrial experiments, often a large number of factors have to be studied, but the experiments are expensive to conduct. In this case, the so called fractional factorial designs, in which only a small fraction of all the possible combinations are observed, are particularly useful. In recent years, factorial designs have received considerable attention, mainly due to the success in applying them to conduct experiments for improving quality and productivity in industrial manufacturing. This research is to study the construction of efficient designs to extract more information, especially when systematic sources of variation (such as heterogeneity of experimental material or day-to-day environmental variations) need to be eliminated to improve the precision.