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）资助的。实验的统计设计和分析是科学发现中有效和常用的工具。技术和计算能力的快速发展使许多复杂的实验成为可能，例如具有分支因子和功能响应的实验。它也带来了许多新的挑战。本提案的主要目标是开发一套新颖有效的统计方法来应对新出现的挑战，从而加速使用实验研究的许多学科的发现。研究计划由两部分组成。研究的第一部分着重于分支因子和嵌套因子实验的设计和分析。在许多复杂的实验中，有些因素只存在于另一个因素的水平之内。这些因素通常被称为嵌套因素。在其中嵌套其他因子的因子称为分支因子。具有分支和嵌套因子的实验设计和分析在许多复杂系统中是至关重要的，但在文献中没有得到太多关注。在本文的第一部分中，提出了新的设计、理论、组合和算法构建策略以及结构化建模，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。研究的第二部分着重于功能响应的计算机实验分析。物理实验既昂贵又耗时；因此，计算机实验作为经济的替代方法被广泛使用。许多计算机实验结果以函数形式收集。然而，由于大多数现有的建模技术都集中在单一输出上，关于模拟功能响应的计算机实验的文献仍然很少。虽然有一些功能响应的降维技术，但它们没有考虑到计算机实验的一个重要特征，即确定性输出。为了解决这个问题，提出了一种时序技术，它提供了一个插值模型。该方法还采用了一种新颖的迭代方法，具有很高的计算效率。本研究中提出的新一类设计、设计理论、组合和算法构建方法以及结构化模型似乎是对分支和嵌套因素实验的第一次系统研究。它们可以为研究问题开辟新的途径，为理论和应用研究注入活力。本文提出的具有功能响应的计算机实验序列建模技术考虑了计算机实验的特殊性，具有较高的计算效率。这是一个创新的概念，可以导致新的研究功能数据分析。这两种方法都很容易适用于各种科学领域，如电子封装、生物力学工程设计、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.