Design and Analysis of Complex Experiments: Branching Factors and Functional Responses

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。实验的统计设计和分析是科学发现中一种有效和常用的工具。技术和计算能力的快速增长使许多复杂的实验成为可能，例如那些具有分支因子和功能反应的实验。这也带来了许多新的挑战。这项提议的主要目标是开发一套新的和有效的统计方法来应对新出现的挑战，从而加速在许多使用实验调查的学科中的发现。研究计划由两部分组成。研究的第一部分主要是设计和分析具有分枝和嵌套因素的实验。在许多复杂的实验中，一些因素只存在于另一个因素的水平内。这类因素通常被称为嵌套因素。嵌套了其他因素的因素称为分支因素。具有分支和嵌套因素的实验的设计和分析在许多复杂系统中是至关重要的，在文献中没有得到太多关注。在本建议的第一部分，提出了新的设计类别、理论、组合和算法构建策略以及结构建模，这些策略可以考虑复杂实验中的分支和嵌套结构，并有效地识别重要因素。本研究的第二部分着重于对具有功能反应的计算机实验进行分析。物理实验可能既昂贵又耗时；因此，计算机实验已被广泛用作经济的替代方案。许多计算机实验响应都是以函数形式收集的。然而，由于现有的大多数建模技术都集中在单个输出上，因此关于用功能反应对计算机实验进行建模的文献仍然很少。虽然有一些功能反应的降维技术，但它们没有考虑到计算机实验的一个重要特征-确定性输出。为了解决这一问题，提出了一种序贯技术，它提供了一种插补模型。它还采用了一种新颖的迭代过程，因此具有很高的计算效率。本研究中提出的新设计类别、设计理论、组合和算法构造方法以及结构模型似乎是第一次对具有分支和嵌套因素的实验进行系统研究。它们可以为研究激发理论和应用研究活力的问题开辟新的途径。本文提出的具有功能反应的计算机实验序贯建模技术考虑了计算机实验的特点，具有较高的计算效率。这是一个创新的概念，可以为函数数据分析带来新的研究。这两种方法都很容易适用于各种科学领域，如电子封装、生物机械工程设计、野火控制和流感建模。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Statistical design and analysis of experiments is an effective and commonly used tool in scientific discoveries. The rapid growth in technology and computing power has made available many complex experiments, such as those with branching factors and functional responses. It also poses many new challenges. The primary objective of this proposal is to develop a set of novel and efficient statistical methods to tackle the emerging challenges and thus accelerate discoveries in many disciplines that use experimental investigation. The research plan consists of two parts. The first part of the research focuses on design and analysis of experiments with branching and nested factors. In many complex experiments, some of the factors exist only within the level of another factor. Such factors are often called nested factors. A factor within which other factors are nested is called a branching factor. Design and analysis of experiments with branching and nested factors are crucial in many complex systems and have not received much attention in the literature. In the first part of this proposal, new classes of designs, theory, combinatorial and algorithmic construction strategies, and structured modeling are proposed that can take into account the branching and nested structure in a complex experiment and identify important factors effectively. The second part of the research focuses on the analysis of computer experiments with functional responses. Physical experiments can be expensive and time-consuming; thus, computer experiments have been widely used as economical alternatives. Many computer experiment responses are collected in a functional form. However, literature on modeling computer experiments with functional responses remains scarce as most of the existing modeling techniques focus on single outputs. Although there are some dimension reduction techniques for functional responses, they do not account for an important feature, the deterministic outputs, of computer experiments. To address this issue, a sequential technique is proposed, which provides an interpolating model. It also incorporates a novel iterative procedure and thus enjoys great computational efficiency.The new class of designs, design theory, combinatorial and algorithmic construction methods, and structured models proposed in this research appears to be the first systematic investigation of experiments with branching and nested factors. They can open new avenues for studying problems that energize both theoretical and applied research. The proposed sequential modeling technique for computer experiments with functional responses takes into account the special features in computer experiments and enjoys great computational efficiency. It is an innovative concept which can lead to new research in functional data analysis. Both methods are readily applicable to a variety of scientific fields, such as electronic packaging, biomechanical engineering design, wildfire control, and influenza modeling.