三水平triple设计的构造与应用

发布时间：2018-01-02 22:02

  本文关键词：三水平triple设计的构造与应用　出处：《吉首大学》2016年硕士论文　论文类型：学位论文

  更多相关文章： 试验设计 double设计 triple设计 示性函数 水平置换 分辨度 最小低阶混杂 均匀设计 可卷L_2-偏差

【摘要】：试验设计是数理统计学中最重要的分支之一,它使研究人员能够找到好的试验有效地进行数据分析.随着科学技术的发展,试验涉及的因素个数众多,且每个因素的水平也较多,此时完全因子设计要求的试验次数大大超过了人们的承受程度.从经济的角度出发,通常采用部分因子设计,它是完全因子设计的一个子集或部分,在实际生活中应用最广泛的是两水平和三水平的部分因子设计.而试验设计的一个重要任务是如何从众多部分因子设计中找到“好”的设计,即用最少的试验次数获得最多的有用信息.对于什么样的设计是“好”的设计,人们基于不同的角度给出了许多设计筛选准则,其中最常见的是分辨度准则和最小广义低阶混杂准则.在构建两水平部分因子设计时一种被称为doubling的方法经常被使用,尤其在构造大型设计时,利用doubling来构造设计既简单又有用,在构造分辨度为ⅣV的设计时,doubling也十分有用,最近已有许多文献讨论了double设计的一些性质和应用.但是,目前的double设计仅限于两水平,于是本文考虑能否把两水平的double设计进行推广,使得构造三水平或者更高水平的大型设计也变得简单有效?同时可以看到两水平double设计的原理一一折叠反转其实是一种非常特殊的水平置换,并且水平置换在设计构造上也有众多应用,因此从水平置换的角度来推广double设计的概念就会有更多选择.但是经过尝试性的探索,可以发现如果行只翻2倍,无论怎么从5种水平置换方式里挑选方案都存在原来的列和新构造的列不会正交,这就意味着所构造的设计性质非常差.但是深入分析,可以发现如果行翻了3倍就会避免这一缺点.因此,本文基于水平置换推广了double设计的概念：提出三水平triple设计的概念.均匀设计及均匀性理论是由我国学者率先提出的一种全新的部分因子设计理论和方法,被广泛用于“计算机仿真试验”和国防、农业、工业、医药和高新技术创新等领域,已取得了显著的经济效益和社会效益.偏差则是衡量均匀设计的测度,常用的偏差有中心化L2-偏差(CD),可卷L2-偏差(W D)，，离散偏差(DD),Lee偏差(LD)等等.基于以上考虑,本论文主要进行了以下3个方面的工作：(1)探索三水平triple设计的形式和结构,基于水平置换构造了三水平的triple设计T(A),并且以示性函数为工具,讨论了T(A)的几个优良性质.(2)讨论了T(A)在可卷L2-偏差下的均匀性：将T(A)的可卷L2-偏差与A的混杂建立了一个等量关系,并且利用泰勒公式给出了T(A)在可卷L2-偏差下紧的下界.(3)讨论了T(A)及其投影设计在构造最小低阶混杂设计上的几个应用.下面简要介绍一下各章的内容.第一章概述了试验设计的相关背景及论文的创新点和结构.第二章简要介绍了基本概念、符号,并给出后面章节要用到的相关结论.第三章讨论了基于水平置换构造了三水平的triple设计T(A),并以示性函数为工具初步分析了triple设计的几个优良性质.如果A的分辨度为Ⅲ,T(A)的分辨度也是Ⅲ.并且进一步地到了T(A)的三类投影设计：如果A的分辨度为Ⅲ或者Ⅳ,那么T(A)的投影设计的分辨度也是Ⅲ或者Ⅳ.第四章深入讨论了T(A)在可卷L2-偏差下均匀性和混杂情况.将T(A)的可卷L2-偏差与A混杂建立了一个等量关系,发现若A有较小的低阶混杂,T(A)会具有较小的可卷L2-偏差和低阶混杂.同时,本章节给出了T(A)在可卷L2-偏差下紧的下界,便于计算机搜索triple设计的均匀设计.第五章讨论了T(A)及其投影设计的几个应用.本章节选取了几个初始设计A,得到的T(A)及其投影设计在可卷L2-偏差下的均匀性都比较好,有的甚至比UD主页中对应的均匀设计还要好,并且他们均具有最小的低阶混杂.第六章对全文工作进行总结和对未来的工作进行了展望.
[Abstract]:The experimental design is one of the most important branches in mathematical statistics, it allows researchers to find a good test effectively for data analysis. With the development of science and technology, a large number of test factors involved, and each factor level is more, the number of test requirements set complete factor greatly exceeds the tolerance level of people. From an economic point of view, usually by fractional factorial design, it is a subset of the full factorial design or part of application in real life is the most widely used part of two level and three level factorial design. An important task of experimental design is how to find a "good" design from many parts factor design, to obtain the most useful information by using the minimal number of tests. For the design of what is "good" design, people from different angles are given based on many design. Standards, is one of the most common resolution criterion and the minimum generalized aberration. In the construction of two level fractional factorial design method called doubling is often used, especially in the construction of large-scale design, using doubling structure design is simple and useful, in the construction of the resolution for the design of IV V when the doubling is also very useful, recently many literatures discussed some properties and applications of double design. However, the current design of the double is limited to the two level, so this paper consider whether to design double two level promotion, the large design structure of the three level or higher level becomes simple and effective? See the principle of two level double design one fold reversal is actually a very special level of replacement, and the level of replacement in the design also has many applications, from the level of replacement The angle to promote the concept of double design will have more choices. But after a tentative exploration, can be found only if the row over 2 times, no matter how columns are not orthogonal from the 5 level replacement selection schemes all have the original columns and new structures, which means that the structure design is made very nature poor. But the in-depth analysis, we can find that if the line over 3 times will avoid the shortcomings. Therefore, this paper generalizes the concept of horizontal displacement based on double design: put forward the concept of three level triple design. Uniform design and uniform theory is proposed by our Ancient Chinese Literature Search took the lead in a new design theory and some factors method, is widely used in "computer simulation" and national defense, industry, agriculture, medicine and other fields of high-tech innovation, has achieved remarkable economic and social benefits. The deviation is a measure of the uniform design 鐨勬祴搴

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