Resolution and Minimum Aberration for Nonregular Factorial Designs

9971212正则分数阶乘是由它的定义关系唯一确定的，它具有简单的混叠结构，即任意两个效果要么正交，要么完全混叠。当我们很少或根本不知道什么影响是潜在的重要影响时，选择具有最小像差的设计是合适的。最小像差设计具有理想的模型鲁棒性，因此可以恰当地称为模型鲁棒设计。正分数阶乘的研究很好，并且在文献中结果丰富。对于非规则的分数阶乘就不能这样说了。不规则分数阶乘由Plackett-Burman设计、Hadamard矩阵或更一般的正交数组给出。规则和非规则设计都允许对所有主要效应进行正交估计。当某些交互具有潜在的重要性时，这两类设计具有相当不同的行为。这使得一些研究者开始研究不规则设计的投影特性。尽管这一重要贡献，还没有一个系统的方法来评估和比较非规则设计。本项目的主要目标是将规则设计中的理论和方法扩展到不规则设计中。j特征的引入促进了这一点，它概括了定义关系的概念。整个项目主要围绕以下几个研究课题进行：(i)激发和引入广义分辨率和最小像差标准，（ii）从设计的可估计性和效率方面研究广义最小像差的隐藏投影特性，（iii）发展j特性理论并建立它们与投影特性的联系，以及（iv）开发、实现和测试用于构建广义最小像差设计的有效计算算法。在许多调查领域，如联邦战略利益领域，有效的数据收集是研究项目最终成功的关键步骤之一。精心策划的实验可确保收集到相关的、翔实的数据。析因设计提供了具有成本效益的实验计划，允许同时有效地研究大量变量，因此广泛用于工业实验，以提高制成品的质量和制造过程的生产率。析因设计可分为两类：规则设计和非规则设计。规则设计得到了很好的研究，并且在文献中有大量的结果。不规则的设计就不是这样了。缺乏通用的理论和方法，以及数据分析的计算困难，是导致非规则设计结果稀少的两个原因。本项目的总体目标是发展研究和构建不规则设计的一般理论和方法。这是通过引入一种称为j特性的工具概念来实现的，该概念能够在投影到较低维度时捕获设计的属性。开发和测试一个用户友好的计算机包是拟议研究的一部分。本研究所发展的理论和方法将为析因设计的研究带来新的启示，为物理、化学和工程科学的实验设计带来新的经济设计，并促进析因设计在生物技术和医学研究等领域的应用，这些领域有巨大的潜力进一步受益于设计方法。

9971212A regular fractional factorial is uniquely determined by its defining relation and it has a simple aliasing structure in that any two effects are either orthogonal or fully aliased. When we have little or no knowledge as for what effects are potentially important, it is appropriate to select designs having minimum aberration. Minimum aberration designs enjoy some desirable model robust properties and therefore can be properly called model robust designs. Regular fractional factorials are well studied and results are abundant in the literature. The same cannot be said of nonregular fractional factorials. Nonregular fractional factorials are given by Plackett-Burman designs, Hadamard matrices, or more generally by orthogonal arrays. Both regular and nonregular designs permit orthogonal estimation of all the main effects. When some interactions are potentially important, the two classes of designs have rather different behaviors. This leads some researchers to study the projection properties of nonregular designs. Despite this important contribution, there has not been a systematic method for assessing and comparing nonregular designs. The broad objective of this project is to extend the theory and methods in regular designs to nonregular designs. This is facilitated by the introduction of J-characteristics, which generalizes the concept of defining relation. The whole project is conducted by focusing on the following research topics: (i) motivate and introduce generalized resolution and minimum aberration criteria, (ii) investigate the hidden projection properties of generalized minimum aberration in terms of estimability and efficiency of designs, (iii) develop a theory of J-characteristics and establish their connections with the projection properties, and (iv) develop, implement, and test efficient computational algorithms for constructing generalized minimum aberration designs.In many areas of investigations, such as those of Federal Strategic Interest, efficient data collection is one of the key steps for the eventual success of a research project. Well planned experiments ensure relevant and informative data to be collected. Factorial designs provide cost-effective experimental plans that allow a large number of variables to be studied simultaneously and efficiently, and are therefore widely used in industrial experiments for improving the quality of manufactured products and the productivity of manufacturing processes. Factorial designs can be categorized into two classes: regular designs and nonregular designs. Regular designs are well studied and results are abundant in the literature. The same cannot be said of nonregular designs. Two reasons for the scarcity of results on nonregular designs are the lack of general theory and methods, and the associated computational difficulties in data analysis. The broad objective of this project is to develop general theory and methods for studying and constructing nonregular designs. This is achieved by introducing an instrumental concept, called J-characteristics, that is capable of capturing the properties of a design when projected onto lower dimensions. Developing and testing a user-friendly computer package is part of the proposed research. The theory and methods to be developed in the proposed research will shed new light on the study of factorial designs, lead to new economical designs for the experiments in the physical, chemical, and engineering sciences, and promote the application of factorial designs in the areas such as biotechnology and medical research that have huge potential to benefit further from the design methodology.