Factorial Designs, An Indicator Function Approach

提案ID：DMS-0306306PI：Kenny Yetile：析因设计，一种指标函数方法摘要：最近的研究表明，指标函数是研究析因设计理论性质的非常有效的工具。指标函数最初由Fontana，Pistone和Rogantin(2000)提出，后来被研究者和他的合作者修改和推广，它为所有的析因设计提供了统一的表示，无论是规则的还是非规则的，两水平的，多水平的或混合水平的。通过将析因设计的指标函数扩展到一个正交多项式基，可以清楚地揭示析因设计的混叠结构。这种方法的一个重要应用是研究正交表的几何结构，这是响应面方法的关键，但过去在很大程度上被忽视了。提出的研究不仅寻求对析因设计的深入理解，而且还寻求发现许多新的有效设计。该项目主要执行三项任务。首先，继续发展指标函数法的理论基础，并将其应用于不同类型的设计，包括区块设计和裂区设计。第二，将最小象差标准推广到所有析因设计。标准应该基于设计的内在结构，而不是基于先验的特定模型。将从理论和经验两方面深入研究与设计效率和估算能力之间的联系。第三，开发并应用序贯算法来构造小规模的非同构正交表的完整目录。析因设计在生命科学和物理科学的研究以及农业和工业研究中都是至关重要的。在调查的筛选和探索阶段，非常规设计比常规设计更有优势。除了应用析因设计的许多传统领域外，将受益的一个特别领域是生物医学研究，在这一领域，探索和筛选调查无处不在，特别是利用微阵列等新技术。许多新的、更有效的设计将在拟议的研究中产生，并通过与科学界其他成员的跨学科合作和与行业的互动来传播。本研究的一个组成部分是在标准实验设计课程的课程中采用指标函数法。这种对现行课程的改变将促进非常规设计在实践中的未来应用。调查人员还致力于让学生参与拟议的研究，特别是那些来自代表性不足群体的学生。此外，这项拟议的研究有望开启代数几何学家和统计学家之间的跨学科交流。

Proposal ID: DMS-0306306PI: Kenny YeTitle: Factorial Designs, An Indicator Function ApproachAbstract:Recent researches show that indicator functions are very effective tools for studying theoretical properties of factorial designs. Initially proposed by Fontana, Pistone and Rogantin (2000) and later modified and generalized by the investigator and his collaborators, indicator functions provide a unified representation for all factorial designs, regular or non-regular, two-level, multi-level or mixed-level. The aliasing structure of a factorial design could be clearly revealed by expanding its indicator function with respect to an orthogonal polynomial basis. An important application of this approach is to study geometric structures of orthogonal arrays, which are essential for response surface methodology but have been largely overlooked in the past. The proposed research seeks not only in depth understanding of factorial designs but also discovery of many new efficient designs. The project pursues three main tasks. First, continue development of theoretical foundation of the indicator function approach and apply it to different types of designs including blocked designs and split-plot designs. Second, generalize minimum aberration criteria to all factorial designs. The criteria should be based on the intrinsic structure of a design but not on a priori specified models. Links to design efficiency and estimation capacity will be thoroughly studied, both theoretically and empirically. Third, develop and apply a sequential algorithm to construct a complete catalogue of non-isomorphic orthogonal arrays of small run sizes.Factorial designs are vital for investigations in both life and physical sciences, as well as in agricultural and industrial studies. Non-regular designs have an advantage over regular designs at the screening and exploration stage of an investigation. Besides many traditional areas that factorial designs are applied, one particular area that will benefit is biomedical research, in which exploration and screening investigations are pervasive, especially with new technologies such as microarrays. Many new, more efficient designs are to be generated in the proposed research and be disseminated through interdisciplinary collaboration with other members of the scientific communities and interactions with industries. An integrated part of the proposed research is to adopt the indicator function approach in the curriculum of standard experimental design courses. Such change from current curriculum will promote the future applications of non-regular designs in practice. The investigator is also committed to involve students in the proposed research, especially those from underrepresented groups. In addition, the proposed research is expected to open interdisciplinary communication between algebraic geometricians and statisticians.