Construction and Analysis of Nonregular and Supersaturated Designs

非规则和超饱和设计通常用于筛选实验，其目的是从大量候选因素中找出几个主导因素。这项建议的目标是构建有效的非规则和过饱和设计，并开发分析此类实验数据的新方法。介绍了利用有限环上的线性码构造非正则设计的一般方法。这些非规则设计在像差、分辨率和投影性方面比文献中已有的设计具有更好的统计性质。利用编码理论中等重码的概念来研究过饱和设计。利用线性规划技术建立了过饱和设计中柱数的新上限。提出了构造有效的非正则过饱和设计的新算法，用于实际应用。提出了一种用于筛选非规则和过饱和设计的有效效应的图解程序和自动程序。仿真结果表明，与文献中已有的方法相比，该方法具有更好的性能，更有效地估计模型的规模。统计设计和实验分析在科学和工业研究开发中得到了广泛的应用。随着科学技术向更高水平发展，研究人员对通过计算机模拟和高性能计算来研究大规模系统越来越感兴趣，并有能力进行研究。两个基本问题是如何设计一个有效的实验和如何正确地分析实验数据。这项拟议的研究旨在开发构建最优设计和分析此类试验的新方法。构建了新的高效、非规则和过饱和的设计，并将在网上提供，以便广泛和快速地传播。建议的研究结果可以很快被吸收到实验设计和分析的研究生课程中。这一建议综合运用了数学和计算工具，并强调了信息论和设计论之间重要的跨学科联系。拟议的研究将在设计理论和数据分析方面带来显著的新进展和更好的实践。

Nonregular and supersaturated designs are commonly used in screening experiments where the goal is to identify a few dominant factors among a large number of candidate factors. The objectives of this proposal are to construct efficient nonregular and supersaturated designs and to develop new methodology for analyzing data from such experiments. A general method is introduced for constructing nonregular designs from linear codes over a finite ring. These nonregular designs have better statistical properties than the existing designs in the literature in terms of aberration, resolution and projectivity. The concept of constant weight codes in coding theory is used for studying supersaturated designs. Linear programming technique is used to establish new upper bounds on the number of columns in a supersaturated design. New algorithms are developed for constructing efficient nonregular and supersaturated designs for practical use. A graphical procedure and an automatic procedure are proposed for screening active effects for nonregular and supersaturated designs. Simulation shows that this method performs well compared to existing methods in the literature and is more efficient at estimating the model size.Statistical design and analysis of experiments have been widely used in scientific and industrial research and development. As science and technology have advanced to a higher level, investigators are becoming more interested in and capable of studying large-scale systems via computer simulations and high-performance computing. Two fundamental questions are how to design an efficient experiment and how to analyze the data properly from the experiment. This proposed research aims at developing novel methods for constructing optimal designs and for analyzing such experiments. New efficient nonregular and supersaturated designs are constructed and will be made available online for broad and quick dissemination. The results of the proposed research can be quickly assimilated into graduate courses on design and analysis of experiments. This proposal employs a combination of mathematical and computational tools and emphasizes an important interdisciplinary connection between information theory and design theory. The proposed research will lead to remarkable new advances in design theory and better practice in data analysis.