Resolution and Minimum Aberration for Nonregular Factorial Designs

9971212正则分数阶乘由它的定义关系唯一确定，它有一个简单的混叠结构，其中任何两个效应要么是正交的，要么是完全混叠的。当我们对潜在的重要影响知之甚少或一无所知时，选择具有最小像差的设计是合适的。最小像差设计具有一些理想的模型稳健性，因此可以恰当地称为模型稳健性设计。正则分数阶乘被研究得很好，在文献中也有丰富的结果。对于非正则分数阶乘就不能这么说了。非正则分数阶乘由Plackett-Burman设计、Hadamard矩阵或更一般的正交表给出。规则设计和非规则设计都允许对所有主效应值进行正交估计。当某些交互可能很重要时，这两类设计具有相当不同的行为。这使得一些研究人员开始研究非规则设计的投影性质。尽管有这些重要的贡献，但还没有一种系统的方法来评估和比较非常规设计。这个项目的主要目标是将规则设计中的理论和方法扩展到非规则设计。这是因为引入了J-特征，它推广了定义关系的概念。整个项目围绕以下研究主题展开：(I)激发和引入广义分辨率和最小像差准则；(Ii)从可估计性和设计效率的角度研究广义最小像差的隐藏投影性质；(Iii)发展J特性理论并建立其与投影性质的联系；(Iv)开发、实现和测试构造广义最小像差设计的高效计算算法。在许多研究领域，如联邦战略利益中心，有效的数据收集是研究项目最终成功的关键步骤之一。精心设计的实验确保收集到相关和信息丰富的数据。析因设计提供了经济有效的实验方案，允许同时有效地研究大量变量，因此被广泛应用于工业实验中，以提高制造产品的质量和制造过程的生产率。析因设计可以分为两类：规则设计和非规则设计。正则设计得到了很好的研究，在文献中也有大量的结果。对于非规则设计则不能说同样的话。非规则设计结果稀少的两个原因是缺乏一般理论和方法，以及数据分析中相关的计算困难。这个项目的主要目标是发展研究和构建非规则设计的一般理论和方法。这是通过引入名为J特征的工具性概念来实现的，该概念能够在投影到较低维度时捕捉设计的属性。开发和测试用户友好的计算机程序包是拟议研究的一部分。本研究拟发展的理论和方法将为析因设计的研究提供新的启示，为物理、化学和工程科学的实验带来新的经济设计，并促进析因设计在生物技术和医学研究等领域的应用，这些领域具有进一步受益于设计方法学的巨大潜力。

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.