模型未知下试验设计的构造
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摘要
试验设计是统计学的重要分支之一,它不仅在理论上有重要意义,在实际领域也具有重大的应用价值。古典的试验设计方法,如正交设计、回归设计、区组设计、拉丁方设计、响应曲面设计等,已建立了丰富的理论。它们均假定模型的形式已知,但其中含有未知参数,要通过试验来估计其中的未知参数,并要求在一定的意义下,试验设计达到最优。这种设计我们称为模型已知下的试验设计。随着新技术和科学的飞速发展,人类面对的问题越来越复杂,需要突破古典试验设计强烈依赖于模型的限制；另外,随着计算机技术的发展,许多试验的前期研究,可以在计算机上进行,这可大大节省试验开支,又可显著加快研究进程。在计算机试验中模型不存在随机试验误差,与模型未知试验的模型(含随机误差)有本质不同.但是在试验设计和建模的要求上,两类试验有许多共性,因此本论文均把它们看作模型未知下的试验
本论文将深入研究模型未知下的试验设计中的一些最新课题.包括：正交拉丁超立方体设计、正交列设计、嵌套空间填充设计、分片空间填充设计以及高水平和混水平因子超饱和设计的构造。另外,本文还将给出产生广义最小低阶混杂设计的一种方法,讨论般高水平因子设计中混杂和均匀性之间的关系,并给出广义最小低阶混杂准则与均匀性的偏差准则的些等价条件。
近年来,基于计算机试验的替代模型(拟模型)在工程设计领域得到了广泛的应用。这主要是因为现实生活中的实际模型往往由复杂的方程表示,这些方程如此复杂以至于只能使用计算机得到近似的数值解。因此,这些方程不但不能准确地描述输入输出变量间的关系,而且计算起来耗时耗力。而拟模型可以在有效识别变量间关系的同时大大地缩短所需的计算时间。
计算机试验没有误差,因此不需要随机化和重复试验,试验中的不确定性只是由于缺乏对输入输出变量间关系的认识而造成的。在计算机试验中应用最广泛的是拉丁超立方体设计,这种一维投影均匀分布的设计是由McKay, Beckman and Conover (1979)首次提出来的。拉丁超立方体设计最初的构造方法是对水平进行随机排序安排因子,因子之间难免具有较高的相关性。对回归模型而言,我们希望变量间是正交的,从而回归系数的估计之间是不相关的。当拟合一阶回归模型时,正交拉丁超立方体设计确保了线性效应估计间的不相关性；进一步当二阶效应,即平方效应和双线性效应存在时,我们还希望线性效应的估计与这些二阶效应的估计是不相关的。因此,我们希望找到有如下性质的拉丁超立方体设计：
(a)因子间的线性效应估计是不相关的；
(b)所有因子的线性效应的估计与所有平方效应和双线性效应的估计是不相关的。现在已有一些具备这两条性质的拉丁超立方体设计,比如Ye(1998)构造的L(2c+1,2c)和L(2c+1+1,2c)；Cioppa and Lucas (2007)通过推广Ye的方法构造的L(2c+1,c+1+(2c))和L(2c+1+1,c+1+(2c)),其中正整数c≤11,Georgiou (2009)通过广义正交设计构造的拉丁超立方体设计。但是,他们所构造的设计相对于试验次数而言所能容纳的因子数太少。另外,由于拉丁超立方体设计要求水平数和试验次数相等,这使得正交拉丁超立方体设计的构造变得非常困难。因此,需要灵活的方法来构造新的正交拉丁超立方体设计,或者是寻找这类设计的替代设计。
为了模拟现实中的系统,人们需要建立数学模型去表示现实的行为。模型可以取成不同的精度,如精细模型和简化模型,来满足试验者多方面的需求。在工程和科学领域,多精度的计算机试验有着广泛的应用。例如,考虑一个只涉及到两个精度的试验,低精度试验和高精度试验,分别用Dl和Dh表示。我们希望构造Dl和Dh满足下面三个原则(Qian. Tang and Wu,2009):
节约：Dh中的试验次数少于Dl的试验次数(这是因为低精度试验比高精度试验便宜);
嵌套关系：Dh嵌套在Dl内,即Dh(?)Dl(为了建模和校正两种试验的差别)；
空间填充设计：Dh和Dl在低维达到分层(假设真模型对设计空间中每个部分都同等看待)。
此外,通常的计算机试验都是建立在假设输入变量是定量因子这一标准框架下的,但是存在一些计算机模型的输入变量是定性的。Qian and Wu (2009)建议用分片空间填充设计去实施既含有定性因子又含有定量因子的计算机试验。显然,分片空间填充设计是一个双精度的试验。这类计算机试验的构造是一个新的课题。
超饱和设计常用于工业及其他科学试验的初期用来筛选活跃因子,在因子数目多而试验次数有限的情况下尤其有用。人们基于不同的考虑提出了很多不同的判断超饱和设计优劣的准则,而基于这些最优准则得到的最优设计可能包含高度非正交的列,甚至完全别名的列。因此,很有必要寻求一种判别列完全别名的简便方法。进一步,如果能构造出任意两列的近似正交性可以被控制的最优超饱和设计,那一定是很有意义的工作。
另外,因子设计的一个最根本的问题是如何从候选设计中挑取一个好的设计。基于各种不同的考虑,涌现出了各种用以选择设计的最优准则,这其中最流行的包括基于广义字长型的广义最小低阶混杂准则和度量均匀性的偏差准则。它们之间有什么关系呢?它们何时等价呢?而且,由于广义字长型是一个向量,所以产生广义最小低阶混杂设计是一个较难的课题。如能给出一种方便产生广义最小低阶混杂设计的方法,将对理论和实际都是非常有用的。
基于以上考虑,本论文主要进行了以下七个方面的工作：
(1)构造具备性质(a)和(b),并且能够包含更多因子的拉丁超立方体设计;
(2)通过放宽因子水平数等于试验次数的要求,构造一类适用于计算机试验的正交列设计；
(3)构造适用于不同精度试验的设计；
(4)构造适合既包含定性因子又包含定量因子的计算机试验的分片空间填充设计。(5)给出判别两列完全别名的等价条件,构造不含完全别名列的一系列最优超饱和设计,并且设计中任意两列的近似正交性是可控的。
(6)给出一种方便产生广义最小低阶混杂设计的方法；
(7)探究均匀性和混杂之间的关系,并讨论均匀性的偏差准则与广义最小低阶混杂准则之间何时等价。
下面简要介绍一下各章的内容。
第一章概述基本概念、符号,并给出后面章节要用到的几个引理。
第二章给出了构造线性效应的估计之间不相关且与二次效应的估计也不相关的正交拉丁超立方体设计的方法。本章完全解决了满足性质(a)和(b)的拉丁超立方体设计L(2c+1,2c)和L(2c+1+1,2c)的构造问题。此种构造方法不但简单易行,而且所构造的设计在所有满足性质(a)和(b)的拉丁超立方体设计中因子数达到最大。同时,我们推广了该构造方法,使其能构造具有更灵活试验次数的拉丁超立方体设计,并且这些设计也满足性质(a)和(b)。在第二章的最后,证明了满足性质(a)和(b)的拉丁超立方体设计在某些给定的准则下也是最优的。
第三章介绍了一类适用于计算机试验的正交列设计的构造方法。就计算机试验而言,虽然拉丁超立方体设计有着广泛的应用,经验表明因子的水平数足够多就好,没必要像拉丁超立方体设计要求的那样一定和试验次数相等,而且拉丁超立方体设计又往往难以正交。本章通过分组旋转正交表中的因子构造了一些适用于计算机试验的正交列设计,用以补充各种因子数和试验数组合的计算机试验的实际需要。这些正交列设计不仅在每个因子上有均匀分布的水平而且在一阶模型中有着不相关的线性效应估计。此外如果所旋转的正交表的强度大于等于三,那么所得设计的线性效应的估计与二次效应的估计之间也是不相关的。由于该种设计的因子数和试验次数之比比较大,所以它也可以应用于筛选因子试验。
第四章给出了一种双精度设计的构造方法。这种设计的构造是个新的课题,因为现有的方法都是致力于构造单精度的试验。Qian, Tang and Wu(2009)认为嵌套空间填充设计适用于这种试验。本章,我们提出了系统构造双精度嵌套空间填充设计的方法,而且这些设计能达到很好的低维分层。
第五章提出了多精度试验设计的构造方法。第四章构造了双精度的嵌套空间填充设计,但是有一些试验需要具有多精度的设计。例如,这些试验可以是物理试验、具体计算机试验和近似计算机试验的组合。本章采用伽罗瓦域分解的方法,产生的设计保证了低维的均匀性,而且这些设计是分片空间填充设计,因此此种设计也可应用于同时包含定性和定量因子的计算机试验。
第六章构造了不含完全别名列的E(fNOD)最优和/或χ2最优的混水平超饱和设计。超饱和设计由于其在因子筛选试验上的显著作用而备受关注。本章给出了两列完全别名的等价条件,并利用等距设计和差阵构造了一系列不含完全别名列的E(fNOD)最优和/或χ2最优的混水平超饱和设计。该方法易于操作,可以构造许多新的不含完全别名列的最优混水平超饱和设计。同时,本章还证明了在列之间的近似正交性方面,所构造设计的任两列的fNOD值可以由初始设计的fNOD值所控制,也就是说如果原设计的fNOD值较小,所得设计的fNOD值也较小。为方便使用,本章还列出了系列新构造的最优混水平超饱和设计。
第七章展示了一种寻找广义最小低阶混杂设计的新算法。因析设计无疑是在工业和科学研究中应用最广泛的试验设计。其成功之处在于能够有效的利用试验同时考查多个因子。最流行的选择因析设计的准则之一是基于广义字长型的广义最小低阶混杂准则。但是寻找具有广义最小低阶混杂的设计是个较难的课题,这是因为该准则具有序贯优化的特点。基于Hickernell and Liu (2002)给出的广义字长型和广义偏差之间的解析表达式,本章把产生广义最小低阶混杂设计的问题转化为约束优化问题,提出了一种解决该问题的有效算法,并构造了一系列的最优设计。
第八章探查了一般多水平因子设计的均匀性和混杂之间的关系。基于广义字长型的广义最小低阶混杂准则和度量均匀性的偏差准则是选取设计的两种重要准则。度量均匀性的偏差准则中离散偏差、中心的L2偏差和可卷的L2偏差应用最为广泛。在本章,我们将这三种偏差表示成了设计的示性函数的二阶多项式,讨论了它们与广义最小字长型之间的密切关系,并给出了这些均匀性准则何时分别与广义最小低阶混杂等价的条件。这些密切关系说明均匀性准则可以用来比较因析设计,同时也说明了采用均匀设计的合理性。所有这些结果都说明正交性和均匀性关系密切,也说明了广义最小低阶混杂标准在均匀性下也是合理的。另外,把离散偏差和可卷的L2偏差表示成示性函数的二次型,有利于寻找相应的最优设计。
Experimental design is an important branch of statistics. It is not only of great theo-retical importance, but also of great value in practical applications. Classical experimental designs, such as orthogonal designs, regression designs, block designs, Latin square designs and response surface designs, have been studied extensively and have rich theories. These designs are all based on known models with unknown parameters and aim at finding the optimal designs and estimating the paramcters. These designs are called experimental designs under known models. With the rapid developments of science and technology, practical problems arc becoming more and more complicated. and the need for breaking through the model depecndence is getting stronger. Besides. with the development of com-puters. more and more experiments can be carried our by means of computer, which can save experimental expense greatly and quicken the pace of research. There is no random error in the computer experiment. which makes it different from the designs under model uncertainty. However. the two kinds of designs have many in common in designing experi-ments and modeling, thus in this dissertation, both of them are regarded as designs under model uncertainty.
This dissertation explores some new subjects of experimental designs under model uncertainty, including the construction of orthogonal Latin hypercube designs (LHDs), orthogonal column designs, nested space-filling designs, sliced space-filling designs and mixed-level supersaturated designs. Besides, a new convenient method of generating gen-eralized minimum aberration (GMA) designs is proposed. Further, we investigate the close relationships among the discrete discrepancy, centered L2-discrepancy (CD2), wrap-around L2-discrepancy (WD2) and generalized wordlength pattern (GWP), and provide some conditions under which a design having one of these minimum discrepancies is equiv-alent to having GMA.
In recent years, building surrogate models (or called metamodels) based on computer experiments has been widely used in engineering field. This is primarily because a physics-based model may be represented by a set of complex equations which often have only numerical solutions that are carried out by computer programs and thus cannot sufficiently describe the relationship between the output and input variables. In addition, it is too time-consuming to solve these complex equations. While a metamodel can effectively identify the input-output relationships and save the computation drastically.
Since there is no random error in a computer experiment, replication and randomiza-tion are in no need and the uncertainty is only due to a lack of knowledge about the nature of the relationship between the inputs and the outputs. Thus computer experiments need special designs. Most commonly used designs in computer experiments are LHDs, which have one dimensional uniformity and were proposed firstly for computer experiments by McKay, Beckman and Conover (1979). However, the original construction of LHDs by mating factors randomly is susceptible to having potential high correlations among fac-tors. It is desirable to include orthogonal variables in a regression model, so that the estimates of the regression coefficients would be uncorrelated. When fitting the first-order model, the orthogonal LHD ensures the independence of estimates of linear effects. Fur-thermore, it is desirable to have an orthogonal LHD that can estimate the linear effects without being correlated with the estimates of quadratic effects and bilinear interactions, when fitting the first-order model while the second-order effects, i.e. the quadratic effects and bilinear interactions, are present. Thus we seek LHDs with the following properties:
(a) the estimates of linear effects of all factors are uncorrelated with each other;
(b) the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions.
There are some existing LHDs with both properties (a) and (b), for example, the L(2c+1,2c) and L(2c+1+1,2c) constructed by Ye (1998); the L(2c+1, c+1+(2c)) and L(2c+1+1, c+ 1+(2c)) for any positive integer c≤11 constructed by Cioppa and Lucas (2007) through extending Ye's procedure; and the LHDs constructed in Georgiou (2009) via (generalized) orthogonal designs, but these designs can accommodate only a few factors. In.addition, the factors in an LHD have as many levels as the run size, which makes it very difficult for an LHD to be orthogonal. Thus, many orthogonal LHDs cannot be constructed by existing methods and even do not exist.
To simulate a physical system, one needs to build mathematical models to represent physical behaviors. Models can take different levels of fidelity such as the detailed and the simple models in order to satisfy the experimenter's various demands. Multi-fidelity com-puter experiments are widely used in many engineering and scientific fields. For example, consider the situation involving two-fidelity experiments that are called the low-accuracy experiment and the high-accuracy experiment. The sets of design points for low-accuracy and high-accuracy are denoted by Dl and Dh, respectively. The two-fidelity computer experiments Dl and Dh are desirable to satisfy the following three principles (Qian, Tang and Wu,2009):
Economy:the number of points in Dh is smaller than the number of points in Dl(low-accuracy experiment is cheaper than high-accuracy experiment);
Nested relationship:Dh is nested within Dl, i.e., Dh(?)Dl(for modeling and calibrating the differences between these two experiments);
Space-filling:both Dh and Dl achieve stratification in low dimensions (suppose the inter-esting features of the true model are as likely to be in one part, of the design space as in another).
On the other hand, the standard framework for computer experiments assumes that the input factors are quantitative, but some input factors of computer models can be qualita-tive. Qian and Wu (2009) proposed the sliced space-filling design for computer experiments with both qualitative and quantitative factors. It is easy to see that the sliced space-filling design is an experiment with two-fidelity. Design construction for such computer experi-ments is a new issue.
Supersaturated design (SSD) is used in the initial stage of an industrial or scientific experiment for screening the active effects, and is useful when there are a large number of factors under investigation while only a very limited number of experimental runs are available. Many optimality criteria have been proposed for design construction and comparison, but the optimal SSDs under these criteria, may include columns with high correlation even fully aliased columns. Thus, it is very necessary to find an easy way to judge whether two columns are fully aliased or not. And it is interesting to construct optimal SSDs with the near-orthogonality between columns being controlled.
In addition, a fundamental and practical question for factorial designs is how to choose a good design from a set of candidates. From different viewpoints, various optimality criteria have been proposed. The GMA based on generalized wordlength pattern (GWP) and uniformity measure of discrepancy are two most popular criteria. What are the relationships among them, when are they equivalent to each other? Besides, the generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. A convenient method for the generation of GMA designs is useful both in theory and practice.
Based on the above discussions, this dissertation is devoted to the following researches:
(1) Constructing LHDs satisfying properties (a) and (b) and can accommodate more fac-tors;
(2) By relaxing the condition that the number of levels for each factor must identical to the run size, constructing a class of orthogonal column designs for computer experiments;
(3) Constructing designs suitable for the experiments with different fidelities;
(4) Constructing sliced space-filling designs for computer experiments with both qualita-tive and quantitative factors;
(5) Providing an equivalent condition for two columns to be fully aliased and constructing a scries of optimal SSDs without fully aliased columns and the near-orthogonality between columns being controlled;
(6) Proposing a convenient method to generate GMA designs;
(7) Exploring the connections between some discrepancies and aberration and showing when they arc equivalent to each other.
In the following, let us introduce the contents of each chapter in brief.
Chapter 1 summarizes some basic concepts, notation and provides some lemmas that will be used in the following chapters.
Chapter 2 is concerned with the construction of LHDs with the properties that the estimates of linear effects of all factors are not only uncorrelated with each other, but also uncorrelated with the estimates of all quadratic effects and bilinear interactions. This chapter completely solves the construction problem for orthogonal LHDs L(2c+1,2c) and L(2c+1+1,2c) with properties (a) and (b). The construction method is convenient and flexible, and the number of factors in any design constructed by the proposed method attains its maximum value among all the corresponding LHDs satisfying properties (a) and (b). Further, we extend the method to construct orthogonal LHDs with more flexible run sizes and properties (a) and (b). At the end of this chapter, we prove that the designs with properties (a) and (b) are optimal under some other meaningful criteria.
Chapter 3 introduces a method for constructing a rich class of orthogonal column designs (OCDs) that are suitable for use in computer experiments. For computer experiments, though LHDs have been popular choices, practical experiences have revealed that designs with enough levels are desirable and it is not essential that the run size equals the numbers of factor levels, as in an LHD which is difficult to be orthogonal. Chapter 3 constructs some new OCDs for computer experiments by rotating groups of factors of orthogonal arrays, which supplement the designs for computer experiments in terms of various run sizes and numbers of factor levels and are flexible in accommodating various combinations of factors with different numbers of levels. These OCDs not only have uniformly spaced levels for each factor but also have independent estimates of the linear effects in a first order model. And the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions if the corresponding orthogonal arrays have strength equal to or greater than three. Along with a large factor-to-run ratio. these new designs arc economical and suitable for screening factors for physical experiments
Chapter 4 presents a method for constructing experiments with two levels of accuracy. Design construction for such computer experiments is a new issue because the existing methods deal almost exclusively with experiments with one level of accuracy, Qian. Tang and Wu (2009) proposed that nested space-filling designs are suitable for conducting such experiments. This chapter proposes a systematic method for constructing nested space-filling designs with two levels of accuracy using the nested difference matrices. These nested space-filling designs can achieve stratification in low dimensions.
In Chapter 5, some methods are proposed for constructing nested space-filling designs with multiple levels of accuracy. Chapter 4 has presented some methods for constructing nested space-filling designs with two levels of accuracy or fidelity. But some experiments can often be run at many different levels of sophistication with vastly varying time, for example, these experiments can be a combination of physical experiment, detailed computer experiment and approximate computer experiment. In this chapter, we propose some methods for constructing nested space-filling designs for multiple experiments with different levels of accuracy. These constructions make use of the decomposition of Galois field and the generated designs achieve uniformity in low dimensions. Further, these deigns are sliced space-filling designs, thus can be used in computer experiments with both quantitative factors and qualitative factors.
Chapter 6 proposes some methods for constructing E(fNOD)- and/orχ2- optimal mixed-level SSDs without fully aliased columns. SSD has received much interest because of its potential in factor screening experiments. In this chapter, we provide an equivalent condition for two columns to be fully aliased and then propose some methods for constructing E(fNOD)-and/orχ2-optimal large mixed-level SSDs without fully aliased columns from small equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs without fully aliased columns can thus be constructed. Further, we prove that the near-orthogonality between columns of the resulting design, measured by fNOD, is well controlled by that of the source designs, i.e., if the source designs have small values of fNOD, then the resulting design tends to have small values of fNOD. In addition, some newly generated optimal mixed-level:SSDs are tabulated for practical use.
In Chapter 7, we provide an algorithmic approach to finding factorial de-signs with GMA. Factorial designs are arguably the most widely used experimental designs in industrial and scientific investigations. Their practical success is due to the efficient use of experimental runs to study many factors simultaneously. One popular cri-terion for selection of fractional factorial designs is GMA. which is based on sequentially minimizing the GWP. The generation of GMA designs is a non-trivial problem due to this sequential optimization nature of the criterion. Based on an analytical expression between the GWP and a general discrepancy provided by Hickernell and Liu (2002), Chapter 6 con-verts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also optimal under the uniformity measure.
Chapter 8 explores the connections between discrepancy and aberration in general multi-level factorials. There are many useful criteria defined from different viewpoints for comparing fractional factorial designs. The uniformity measure of discrep-ancy and GMA are two popular and important ones. Among various existing discrepancies, the discrete discrepancy, centered L2-discrepancy(CD2) and wrap-around L2-discrepancy (WD2) have been well justified and widely used. In chapter 8, we express the discrete dis-crepancy, CD2 and WD2 in the second-order polynomials of the indicator functions, and then investigate the relationships between these discrepancies and the GWP. Further, we provide some conditions under which a design having one of these minimum discrepancies is equivalent to having GMA. The close relationships among these criteria further show that the uniformity criteria can be utilized to compare fractional factorial designs and provide an additional rationale for employing uniform designs. All these results show that orthogonality is strongly related to uniformity, and provide some further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrete discrepancy and WD2 in the quadratic forms of the indicator functions are useful for us to find optimal designs under each of the criteria.
本论文将深入研究模型未知下的试验设计中的一些最新课题.包括：正交拉丁超立方体设计、正交列设计、嵌套空间填充设计、分片空间填充设计以及高水平和混水平因子超饱和设计的构造。另外,本文还将给出产生广义最小低阶混杂设计的一种方法,讨论般高水平因子设计中混杂和均匀性之间的关系,并给出广义最小低阶混杂准则与均匀性的偏差准则的些等价条件。
近年来,基于计算机试验的替代模型(拟模型)在工程设计领域得到了广泛的应用。这主要是因为现实生活中的实际模型往往由复杂的方程表示,这些方程如此复杂以至于只能使用计算机得到近似的数值解。因此,这些方程不但不能准确地描述输入输出变量间的关系,而且计算起来耗时耗力。而拟模型可以在有效识别变量间关系的同时大大地缩短所需的计算时间。
计算机试验没有误差,因此不需要随机化和重复试验,试验中的不确定性只是由于缺乏对输入输出变量间关系的认识而造成的。在计算机试验中应用最广泛的是拉丁超立方体设计,这种一维投影均匀分布的设计是由McKay, Beckman and Conover (1979)首次提出来的。拉丁超立方体设计最初的构造方法是对水平进行随机排序安排因子,因子之间难免具有较高的相关性。对回归模型而言,我们希望变量间是正交的,从而回归系数的估计之间是不相关的。当拟合一阶回归模型时,正交拉丁超立方体设计确保了线性效应估计间的不相关性；进一步当二阶效应,即平方效应和双线性效应存在时,我们还希望线性效应的估计与这些二阶效应的估计是不相关的。因此,我们希望找到有如下性质的拉丁超立方体设计：
(a)因子间的线性效应估计是不相关的；
(b)所有因子的线性效应的估计与所有平方效应和双线性效应的估计是不相关的。现在已有一些具备这两条性质的拉丁超立方体设计,比如Ye(1998)构造的L(2c+1,2c)和L(2c+1+1,2c)；Cioppa and Lucas (2007)通过推广Ye的方法构造的L(2c+1,c+1+(2c))和L(2c+1+1,c+1+(2c)),其中正整数c≤11,Georgiou (2009)通过广义正交设计构造的拉丁超立方体设计。但是,他们所构造的设计相对于试验次数而言所能容纳的因子数太少。另外,由于拉丁超立方体设计要求水平数和试验次数相等,这使得正交拉丁超立方体设计的构造变得非常困难。因此,需要灵活的方法来构造新的正交拉丁超立方体设计,或者是寻找这类设计的替代设计。
为了模拟现实中的系统,人们需要建立数学模型去表示现实的行为。模型可以取成不同的精度,如精细模型和简化模型,来满足试验者多方面的需求。在工程和科学领域,多精度的计算机试验有着广泛的应用。例如,考虑一个只涉及到两个精度的试验,低精度试验和高精度试验,分别用Dl和Dh表示。我们希望构造Dl和Dh满足下面三个原则(Qian. Tang and Wu,2009):
节约：Dh中的试验次数少于Dl的试验次数(这是因为低精度试验比高精度试验便宜);
嵌套关系：Dh嵌套在Dl内,即Dh(?)Dl(为了建模和校正两种试验的差别)；
空间填充设计：Dh和Dl在低维达到分层(假设真模型对设计空间中每个部分都同等看待)。
此外,通常的计算机试验都是建立在假设输入变量是定量因子这一标准框架下的,但是存在一些计算机模型的输入变量是定性的。Qian and Wu (2009)建议用分片空间填充设计去实施既含有定性因子又含有定量因子的计算机试验。显然,分片空间填充设计是一个双精度的试验。这类计算机试验的构造是一个新的课题。
超饱和设计常用于工业及其他科学试验的初期用来筛选活跃因子,在因子数目多而试验次数有限的情况下尤其有用。人们基于不同的考虑提出了很多不同的判断超饱和设计优劣的准则,而基于这些最优准则得到的最优设计可能包含高度非正交的列,甚至完全别名的列。因此,很有必要寻求一种判别列完全别名的简便方法。进一步,如果能构造出任意两列的近似正交性可以被控制的最优超饱和设计,那一定是很有意义的工作。
另外,因子设计的一个最根本的问题是如何从候选设计中挑取一个好的设计。基于各种不同的考虑,涌现出了各种用以选择设计的最优准则,这其中最流行的包括基于广义字长型的广义最小低阶混杂准则和度量均匀性的偏差准则。它们之间有什么关系呢?它们何时等价呢?而且,由于广义字长型是一个向量,所以产生广义最小低阶混杂设计是一个较难的课题。如能给出一种方便产生广义最小低阶混杂设计的方法,将对理论和实际都是非常有用的。
基于以上考虑,本论文主要进行了以下七个方面的工作：
(1)构造具备性质(a)和(b),并且能够包含更多因子的拉丁超立方体设计;
(2)通过放宽因子水平数等于试验次数的要求,构造一类适用于计算机试验的正交列设计；
(3)构造适用于不同精度试验的设计；
(4)构造适合既包含定性因子又包含定量因子的计算机试验的分片空间填充设计。(5)给出判别两列完全别名的等价条件,构造不含完全别名列的一系列最优超饱和设计,并且设计中任意两列的近似正交性是可控的。
(6)给出一种方便产生广义最小低阶混杂设计的方法；
(7)探究均匀性和混杂之间的关系,并讨论均匀性的偏差准则与广义最小低阶混杂准则之间何时等价。
下面简要介绍一下各章的内容。
第一章概述基本概念、符号,并给出后面章节要用到的几个引理。
第二章给出了构造线性效应的估计之间不相关且与二次效应的估计也不相关的正交拉丁超立方体设计的方法。本章完全解决了满足性质(a)和(b)的拉丁超立方体设计L(2c+1,2c)和L(2c+1+1,2c)的构造问题。此种构造方法不但简单易行,而且所构造的设计在所有满足性质(a)和(b)的拉丁超立方体设计中因子数达到最大。同时,我们推广了该构造方法,使其能构造具有更灵活试验次数的拉丁超立方体设计,并且这些设计也满足性质(a)和(b)。在第二章的最后,证明了满足性质(a)和(b)的拉丁超立方体设计在某些给定的准则下也是最优的。
第三章介绍了一类适用于计算机试验的正交列设计的构造方法。就计算机试验而言,虽然拉丁超立方体设计有着广泛的应用,经验表明因子的水平数足够多就好,没必要像拉丁超立方体设计要求的那样一定和试验次数相等,而且拉丁超立方体设计又往往难以正交。本章通过分组旋转正交表中的因子构造了一些适用于计算机试验的正交列设计,用以补充各种因子数和试验数组合的计算机试验的实际需要。这些正交列设计不仅在每个因子上有均匀分布的水平而且在一阶模型中有着不相关的线性效应估计。此外如果所旋转的正交表的强度大于等于三,那么所得设计的线性效应的估计与二次效应的估计之间也是不相关的。由于该种设计的因子数和试验次数之比比较大,所以它也可以应用于筛选因子试验。
第四章给出了一种双精度设计的构造方法。这种设计的构造是个新的课题,因为现有的方法都是致力于构造单精度的试验。Qian, Tang and Wu(2009)认为嵌套空间填充设计适用于这种试验。本章,我们提出了系统构造双精度嵌套空间填充设计的方法,而且这些设计能达到很好的低维分层。
第五章提出了多精度试验设计的构造方法。第四章构造了双精度的嵌套空间填充设计,但是有一些试验需要具有多精度的设计。例如,这些试验可以是物理试验、具体计算机试验和近似计算机试验的组合。本章采用伽罗瓦域分解的方法,产生的设计保证了低维的均匀性,而且这些设计是分片空间填充设计,因此此种设计也可应用于同时包含定性和定量因子的计算机试验。
第六章构造了不含完全别名列的E(fNOD)最优和/或χ2最优的混水平超饱和设计。超饱和设计由于其在因子筛选试验上的显著作用而备受关注。本章给出了两列完全别名的等价条件,并利用等距设计和差阵构造了一系列不含完全别名列的E(fNOD)最优和/或χ2最优的混水平超饱和设计。该方法易于操作,可以构造许多新的不含完全别名列的最优混水平超饱和设计。同时,本章还证明了在列之间的近似正交性方面,所构造设计的任两列的fNOD值可以由初始设计的fNOD值所控制,也就是说如果原设计的fNOD值较小,所得设计的fNOD值也较小。为方便使用,本章还列出了系列新构造的最优混水平超饱和设计。
第七章展示了一种寻找广义最小低阶混杂设计的新算法。因析设计无疑是在工业和科学研究中应用最广泛的试验设计。其成功之处在于能够有效的利用试验同时考查多个因子。最流行的选择因析设计的准则之一是基于广义字长型的广义最小低阶混杂准则。但是寻找具有广义最小低阶混杂的设计是个较难的课题,这是因为该准则具有序贯优化的特点。基于Hickernell and Liu (2002)给出的广义字长型和广义偏差之间的解析表达式,本章把产生广义最小低阶混杂设计的问题转化为约束优化问题,提出了一种解决该问题的有效算法,并构造了一系列的最优设计。
第八章探查了一般多水平因子设计的均匀性和混杂之间的关系。基于广义字长型的广义最小低阶混杂准则和度量均匀性的偏差准则是选取设计的两种重要准则。度量均匀性的偏差准则中离散偏差、中心的L2偏差和可卷的L2偏差应用最为广泛。在本章,我们将这三种偏差表示成了设计的示性函数的二阶多项式,讨论了它们与广义最小字长型之间的密切关系,并给出了这些均匀性准则何时分别与广义最小低阶混杂等价的条件。这些密切关系说明均匀性准则可以用来比较因析设计,同时也说明了采用均匀设计的合理性。所有这些结果都说明正交性和均匀性关系密切,也说明了广义最小低阶混杂标准在均匀性下也是合理的。另外,把离散偏差和可卷的L2偏差表示成示性函数的二次型,有利于寻找相应的最优设计。
Experimental design is an important branch of statistics. It is not only of great theo-retical importance, but also of great value in practical applications. Classical experimental designs, such as orthogonal designs, regression designs, block designs, Latin square designs and response surface designs, have been studied extensively and have rich theories. These designs are all based on known models with unknown parameters and aim at finding the optimal designs and estimating the paramcters. These designs are called experimental designs under known models. With the rapid developments of science and technology, practical problems arc becoming more and more complicated. and the need for breaking through the model depecndence is getting stronger. Besides. with the development of com-puters. more and more experiments can be carried our by means of computer, which can save experimental expense greatly and quicken the pace of research. There is no random error in the computer experiment. which makes it different from the designs under model uncertainty. However. the two kinds of designs have many in common in designing experi-ments and modeling, thus in this dissertation, both of them are regarded as designs under model uncertainty.
This dissertation explores some new subjects of experimental designs under model uncertainty, including the construction of orthogonal Latin hypercube designs (LHDs), orthogonal column designs, nested space-filling designs, sliced space-filling designs and mixed-level supersaturated designs. Besides, a new convenient method of generating gen-eralized minimum aberration (GMA) designs is proposed. Further, we investigate the close relationships among the discrete discrepancy, centered L2-discrepancy (CD2), wrap-around L2-discrepancy (WD2) and generalized wordlength pattern (GWP), and provide some conditions under which a design having one of these minimum discrepancies is equiv-alent to having GMA.
In recent years, building surrogate models (or called metamodels) based on computer experiments has been widely used in engineering field. This is primarily because a physics-based model may be represented by a set of complex equations which often have only numerical solutions that are carried out by computer programs and thus cannot sufficiently describe the relationship between the output and input variables. In addition, it is too time-consuming to solve these complex equations. While a metamodel can effectively identify the input-output relationships and save the computation drastically.
Since there is no random error in a computer experiment, replication and randomiza-tion are in no need and the uncertainty is only due to a lack of knowledge about the nature of the relationship between the inputs and the outputs. Thus computer experiments need special designs. Most commonly used designs in computer experiments are LHDs, which have one dimensional uniformity and were proposed firstly for computer experiments by McKay, Beckman and Conover (1979). However, the original construction of LHDs by mating factors randomly is susceptible to having potential high correlations among fac-tors. It is desirable to include orthogonal variables in a regression model, so that the estimates of the regression coefficients would be uncorrelated. When fitting the first-order model, the orthogonal LHD ensures the independence of estimates of linear effects. Fur-thermore, it is desirable to have an orthogonal LHD that can estimate the linear effects without being correlated with the estimates of quadratic effects and bilinear interactions, when fitting the first-order model while the second-order effects, i.e. the quadratic effects and bilinear interactions, are present. Thus we seek LHDs with the following properties:
(a) the estimates of linear effects of all factors are uncorrelated with each other;
(b) the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions.
There are some existing LHDs with both properties (a) and (b), for example, the L(2c+1,2c) and L(2c+1+1,2c) constructed by Ye (1998); the L(2c+1, c+1+(2c)) and L(2c+1+1, c+ 1+(2c)) for any positive integer c≤11 constructed by Cioppa and Lucas (2007) through extending Ye's procedure; and the LHDs constructed in Georgiou (2009) via (generalized) orthogonal designs, but these designs can accommodate only a few factors. In.addition, the factors in an LHD have as many levels as the run size, which makes it very difficult for an LHD to be orthogonal. Thus, many orthogonal LHDs cannot be constructed by existing methods and even do not exist.
To simulate a physical system, one needs to build mathematical models to represent physical behaviors. Models can take different levels of fidelity such as the detailed and the simple models in order to satisfy the experimenter's various demands. Multi-fidelity com-puter experiments are widely used in many engineering and scientific fields. For example, consider the situation involving two-fidelity experiments that are called the low-accuracy experiment and the high-accuracy experiment. The sets of design points for low-accuracy and high-accuracy are denoted by Dl and Dh, respectively. The two-fidelity computer experiments Dl and Dh are desirable to satisfy the following three principles (Qian, Tang and Wu,2009):
Economy:the number of points in Dh is smaller than the number of points in Dl(low-accuracy experiment is cheaper than high-accuracy experiment);
Nested relationship:Dh is nested within Dl, i.e., Dh(?)Dl(for modeling and calibrating the differences between these two experiments);
Space-filling:both Dh and Dl achieve stratification in low dimensions (suppose the inter-esting features of the true model are as likely to be in one part, of the design space as in another).
On the other hand, the standard framework for computer experiments assumes that the input factors are quantitative, but some input factors of computer models can be qualita-tive. Qian and Wu (2009) proposed the sliced space-filling design for computer experiments with both qualitative and quantitative factors. It is easy to see that the sliced space-filling design is an experiment with two-fidelity. Design construction for such computer experi-ments is a new issue.
Supersaturated design (SSD) is used in the initial stage of an industrial or scientific experiment for screening the active effects, and is useful when there are a large number of factors under investigation while only a very limited number of experimental runs are available. Many optimality criteria have been proposed for design construction and comparison, but the optimal SSDs under these criteria, may include columns with high correlation even fully aliased columns. Thus, it is very necessary to find an easy way to judge whether two columns are fully aliased or not. And it is interesting to construct optimal SSDs with the near-orthogonality between columns being controlled.
In addition, a fundamental and practical question for factorial designs is how to choose a good design from a set of candidates. From different viewpoints, various optimality criteria have been proposed. The GMA based on generalized wordlength pattern (GWP) and uniformity measure of discrepancy are two most popular criteria. What are the relationships among them, when are they equivalent to each other? Besides, the generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. A convenient method for the generation of GMA designs is useful both in theory and practice.
Based on the above discussions, this dissertation is devoted to the following researches:
(1) Constructing LHDs satisfying properties (a) and (b) and can accommodate more fac-tors;
(2) By relaxing the condition that the number of levels for each factor must identical to the run size, constructing a class of orthogonal column designs for computer experiments;
(3) Constructing designs suitable for the experiments with different fidelities;
(4) Constructing sliced space-filling designs for computer experiments with both qualita-tive and quantitative factors;
(5) Providing an equivalent condition for two columns to be fully aliased and constructing a scries of optimal SSDs without fully aliased columns and the near-orthogonality between columns being controlled;
(6) Proposing a convenient method to generate GMA designs;
(7) Exploring the connections between some discrepancies and aberration and showing when they arc equivalent to each other.
In the following, let us introduce the contents of each chapter in brief.
Chapter 1 summarizes some basic concepts, notation and provides some lemmas that will be used in the following chapters.
Chapter 2 is concerned with the construction of LHDs with the properties that the estimates of linear effects of all factors are not only uncorrelated with each other, but also uncorrelated with the estimates of all quadratic effects and bilinear interactions. This chapter completely solves the construction problem for orthogonal LHDs L(2c+1,2c) and L(2c+1+1,2c) with properties (a) and (b). The construction method is convenient and flexible, and the number of factors in any design constructed by the proposed method attains its maximum value among all the corresponding LHDs satisfying properties (a) and (b). Further, we extend the method to construct orthogonal LHDs with more flexible run sizes and properties (a) and (b). At the end of this chapter, we prove that the designs with properties (a) and (b) are optimal under some other meaningful criteria.
Chapter 3 introduces a method for constructing a rich class of orthogonal column designs (OCDs) that are suitable for use in computer experiments. For computer experiments, though LHDs have been popular choices, practical experiences have revealed that designs with enough levels are desirable and it is not essential that the run size equals the numbers of factor levels, as in an LHD which is difficult to be orthogonal. Chapter 3 constructs some new OCDs for computer experiments by rotating groups of factors of orthogonal arrays, which supplement the designs for computer experiments in terms of various run sizes and numbers of factor levels and are flexible in accommodating various combinations of factors with different numbers of levels. These OCDs not only have uniformly spaced levels for each factor but also have independent estimates of the linear effects in a first order model. And the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions if the corresponding orthogonal arrays have strength equal to or greater than three. Along with a large factor-to-run ratio. these new designs arc economical and suitable for screening factors for physical experiments
Chapter 4 presents a method for constructing experiments with two levels of accuracy. Design construction for such computer experiments is a new issue because the existing methods deal almost exclusively with experiments with one level of accuracy, Qian. Tang and Wu (2009) proposed that nested space-filling designs are suitable for conducting such experiments. This chapter proposes a systematic method for constructing nested space-filling designs with two levels of accuracy using the nested difference matrices. These nested space-filling designs can achieve stratification in low dimensions.
In Chapter 5, some methods are proposed for constructing nested space-filling designs with multiple levels of accuracy. Chapter 4 has presented some methods for constructing nested space-filling designs with two levels of accuracy or fidelity. But some experiments can often be run at many different levels of sophistication with vastly varying time, for example, these experiments can be a combination of physical experiment, detailed computer experiment and approximate computer experiment. In this chapter, we propose some methods for constructing nested space-filling designs for multiple experiments with different levels of accuracy. These constructions make use of the decomposition of Galois field and the generated designs achieve uniformity in low dimensions. Further, these deigns are sliced space-filling designs, thus can be used in computer experiments with both quantitative factors and qualitative factors.
Chapter 6 proposes some methods for constructing E(fNOD)- and/orχ2- optimal mixed-level SSDs without fully aliased columns. SSD has received much interest because of its potential in factor screening experiments. In this chapter, we provide an equivalent condition for two columns to be fully aliased and then propose some methods for constructing E(fNOD)-and/orχ2-optimal large mixed-level SSDs without fully aliased columns from small equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs without fully aliased columns can thus be constructed. Further, we prove that the near-orthogonality between columns of the resulting design, measured by fNOD, is well controlled by that of the source designs, i.e., if the source designs have small values of fNOD, then the resulting design tends to have small values of fNOD. In addition, some newly generated optimal mixed-level:SSDs are tabulated for practical use.
In Chapter 7, we provide an algorithmic approach to finding factorial de-signs with GMA. Factorial designs are arguably the most widely used experimental designs in industrial and scientific investigations. Their practical success is due to the efficient use of experimental runs to study many factors simultaneously. One popular cri-terion for selection of fractional factorial designs is GMA. which is based on sequentially minimizing the GWP. The generation of GMA designs is a non-trivial problem due to this sequential optimization nature of the criterion. Based on an analytical expression between the GWP and a general discrepancy provided by Hickernell and Liu (2002), Chapter 6 con-verts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also optimal under the uniformity measure.
Chapter 8 explores the connections between discrepancy and aberration in general multi-level factorials. There are many useful criteria defined from different viewpoints for comparing fractional factorial designs. The uniformity measure of discrep-ancy and GMA are two popular and important ones. Among various existing discrepancies, the discrete discrepancy, centered L2-discrepancy(CD2) and wrap-around L2-discrepancy (WD2) have been well justified and widely used. In chapter 8, we express the discrete dis-crepancy, CD2 and WD2 in the second-order polynomials of the indicator functions, and then investigate the relationships between these discrepancies and the GWP. Further, we provide some conditions under which a design having one of these minimum discrepancies is equivalent to having GMA. The close relationships among these criteria further show that the uniformity criteria can be utilized to compare fractional factorial designs and provide an additional rationale for employing uniform designs. All these results show that orthogonality is strongly related to uniformity, and provide some further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrete discrepancy and WD2 in the quadratic forms of the indicator functions are useful for us to find optimal designs under each of the criteria.
引文
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