Construction and Analysis of Nonregular and Supersaturated Designs

不规则和过饱和设计通常用于筛选实验，其目标是在大量候选因素中确定少数主导因素。本提案的目标是建立有效的不规则和过饱和设计，并开发新的方法来分析这些实验的数据。介绍了用有限环上的线性码构造非正则设计的一般方法。这些不规则设计在像差、分辨率和投影性方面都比文献中已有的设计有更好的统计性能。用编码理论中的等权码概念来研究过饱和设计。在过饱和设计中，采用线性规划技术建立新的列数上界。开发了新的算法来构造有效的非规则和过饱和设计，以供实际使用。提出了一种图形程序和一种自动程序来筛选不规则和过饱和设计的主动效应。仿真结果表明，与文献中已有的方法相比，该方法性能较好，在模型大小估计方面效率更高。实验的统计设计和分析已广泛应用于科学和工业研究与开发。随着科学技术的发展，研究者们对通过计算机模拟和高性能计算来研究大型系统越来越感兴趣，也越来越有能力。两个基本问题是如何设计一个有效的实验和如何从实验中正确地分析数据。本研究旨在开发新的方法来构建最优设计和分析这些实验。新的高效的不规则和过饱和设计将被构建，并将在网上提供，以广泛和快速传播。所提出的研究结果可以很快地融入到研究生的实验设计和分析课程中。该建议采用数学和计算工具的结合，并强调信息理论和设计理论之间重要的跨学科联系。所提出的研究将导致设计理论的显著新进展和更好的数据分析实践。

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.