因析设计中几类问题的研究与应用

发布时间：2018-07-02 19:42

本文选题：因析设计 + 混合水平　；参考：《东南大学》2015年博士论文

【摘要】：本文主要研究了因析设计中的几类典型问题,包括利用矩阵象理论研究了非正规因析设计的混杂度量及其在全局敏感性分析中的应用、利用矩阵象工具建立了广义分辨度指标、建立了广义可变分辨度设计的一般理论及相应设计的构造方法、完整讨论了4类部分纯净主效应设计的存在性和构造、给出了通过区组因子达到正交的主效应设计的若干构造方法、及其在均匀性准则下的最优设计.主要内容如下：第一章着重介绍试验设计的基本概念以及重要理论,介绍了因析设计、因子分组设计的研究进展.第一章的结尾处,大致陈述了本文的主要工作,并阐释了创新点.第二章我们通过运用矩阵象的相关性质,提出了一种区分带有复杂混杂结构的非正规设计的混杂度量工具,其可以适用于正规设计,也可以适用于非正规设计,实例分析表明该准则相比于几种经典方法有更强的区分能力.另一方面,本章还考虑了当所采用正交表强度小于所需强度时,对全局敏感性分析的影响.基于矩阵象理论,我们首先推广了方差分析高维模型下的别名矩阵,然后通过依次最小化平方混杂度,给出了一种敏感性指标的估计方法.讨论了用二水平16次设计和四水平64次设计来估计低阶显著性敏感性指标和高阶敏感性指标的例子.所有例子表明当设计的平方混杂度越小时,其全局敏感性指标的估计具有更小的偏差和方差.第三章讨论当组内因子间存在不可忽略交互效应时的广义可变分辨度设计D(n, (m1, m2),(τ1,τ2)τ3,τ).我们讨论了广义可变分辨度设计的存在性条件,并给出了与折衷设计、纯净折衷设计、带有部分纯净二因子交互作用设计之间的关系.最后我们给出了广义可变分辨度设计的构造方法.第四章论证了广义可变分辨度设计在方差分析高维模型下估计参数时具有A最优性.模型分别考虑了不含交互作用和含有交互作用两种情形.最后也通过模拟进行了验证比较.第五章研究了部分纯净主效应设计.纯净效应准则是选择最优设计的一个重要原则.在一个分辨度为V的设计中,所有主效应是纯净的.但如果还有信息表明部分二阶交互效应是不存在的,则可以得到一类新的设计,其所含有的主效应与可能存在的二阶交互效应均正交.且这类设计的列数可以大于分辨度V的设计列数.我们称其为部分纯净主效应设计.本章完整的讨论了所有可能的部分纯净主效应设计.并对这4类设计的存在性进行了研究,最后发现其中3类是不存在的.对可能存在的第4类设计给出了相应的构造方法.第六章研究了通过区组因子达到正交的主效应设计(POTB)的若干构造方法.这类设计的处理因子能够通过区组因子达到两两正交.然而,文献中的构造方法较少.本章我们提出若干构造试验次数较小,多水平的,饱和POTB设计的方法,且这些设计均是可连接的和方差平衡的.第七章讨论了上述设计(通过区组设计得到正交的主效应设计)中的正交性可以在水平置换下保持不变.然而,水平置换可以改变设计的几何结构和统计性质.于是本章进一步采用均匀性来区分同一参数下的POTB设计.通过本章提出的最优化算法可以得到很多最优的或者近似最优的均匀POTB设计.
[Abstract]:This paper mainly studies several typical problems in the factorial design, including the use of matrix image theory to study the hybrid measurement of the irregular factorial design and its application in the global sensitivity analysis. The generalized resolution index is established by the matrix image tool, and the general theory of the generalized variable discrimination design and the corresponding design are established. In this method, the existence and construction of the 4 kinds of pure main effect design are discussed, and some construction methods of the main effect design through the region group factor are given and the optimal design under the uniformity criterion are given. The main contents are as follows: the first chapter introduces the basic concepts and important theories of the experimental design, and introduces the basic concepts. In the end of the first chapter, the main work of this paper is roughly stated and the innovation point is explained. In the second chapter, by using the related properties of matrix images, we propose a hybrid metric tool for irregular design with complex hybrid structures, which can be applied to regular design, It can be applied to irregular design. Example analysis shows that the criterion has a better distinction than several classical methods. On the other hand, this chapter also considers the influence on the global sensitivity analysis when the strength of the orthogonal table is less than the required strength. Based on the matrix image theory, we first generalize the high dimensional model of variance analysis. The alias matrix, and then an estimation method of sensitivity index is given by minimizing the square miscellaneous degree in turn. An example of estimating low order saliency sensitivity index and high order sensitivity index with two level 16 times design and four level 64 times design is discussed. All examples show that the overall sensitivity of the design is the hourly mixed degree of design. The estimation of the perceptual index has smaller deviations and variance. In the third chapter, we discuss the generalized variable resolution design D (n, (M1, M2), (tau, tau 2) tau 3, tau (tau) when there is no neglecting the interaction effect among the factors in the group. We discuss the existence condition of the generalized variable resolution design, and give the tradeoff design, the pure compromise design, with some part. The relationship between the interaction design of the pure two factor interaction. Finally, we give the construction method of the generalized variable resolution design. The fourth chapter demonstrates that the generalized variable resolution design has the A optimality in the estimation of the parameters under the high dimensional model of variance analysis. The model considers two cases without interaction and interaction, respectively. In the fifth chapter, the fifth chapter studies the design of the partial pure main effect. The pure effect criterion is an important principle for the selection of optimal design. In a design with a resolution of V, all the main effects are pure. But if there is any information that the part of the two order interaction effect does not exist, we can get a new kind of new type. The main effect contained in the design is orthogonal to the possible two order interaction effects. And the number of columns in this type of design can be larger than the number of design columns of the resolution V. We call it a part of the pure main effect design. This chapter completely discusses all possible parts of the pure main effect design. And the existence of these 4 kinds of design is studied. Finally, we find that the 3 types are nonexistent. The corresponding construction methods are given for the possible fourth types of design. The sixth chapter studies several construction methods of the main effect design (POTB) through the region group factor. The processing factor of this kind of design can reach 22 orthogonal by the region group factor. However, the construction method in the literature is more than that in the literature. In this chapter, we propose a number of smaller, multi horizontal, saturated POTB design methods, and these designs are both connectable and variance balanced. The seventh chapter discusses that the above design (through the design of the orthogonal main effect design by the area group design) can remain unchanged under the horizontal displacement. However, the horizontal displacement can be maintained. In order to change the geometric structure and statistical properties of the design, this chapter further uses uniformity to distinguish the POTB design under the same parameter. The optimization algorithm proposed in this chapter can obtain many optimal or approximately optimal uniform POTB designs.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TB47

【参考文献】