拉丁超立方体设计的构造与超饱和设计的分析
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摘要
科学试验是人们认识自然,了解自然的重要手段,它被广泛应用于人类生活实践的各个方面。随着科学技术的发展,试验研究的对象涉及因素越来越多,因素之间的关系越来复杂,仅靠直觉和经验已经远远不能满足试验要求,于是试验设计应运而生。设计一个试验要考虑的环节很多：试验目的、试验方案、试验设计、试验实施、数据分析,其中需要利用数学和统计学知识来解决的是试验方案的设计和数据分析两大环节。试验的设计问题和分析问题之间是有制约关系的,设计出的试验不仅要求试验次数尽量少,而且所包含的有用信息要尽量多,要便于试验结果的分析：分析方法又要依赖于设计方法,不同的设计方法的试验结果需要采用不同的分析方法。
试验设计源于统计学家R. A. Fisher在上世纪30年代英国Rothamsted农场试验站的开创性工作,至今已有80多年历史。最初的试验设计问题都是受农业和生物学中的问题的激励,像著名的孟德尔豌豆试验。这些试验的特点是试验结果受随机误差的影响,同样的条件,其观测结果会因随机误差而产生波动。在试验室、工厂或者农场中进行的实体试验,都不可避免的具有这种随机性。我们称这样的试验为实体试验。针对实体试验,提出了很多的设计准则：随机化、分区组、重复性、正交性等。基于这些准则的设计,像正交设计、回归设计、区组设计和拉丁方设计,在实践中已经得到成功的应用。近几十年来,随着计算机技术的飞速发展,计算机在研究中得到越来越多的应用。传统的实体试验存在花费高、周期长等问题,甚至一些试验根本不可能进行,像研究飓风破会性问题,这时在计算机上进行模拟试验成为一个新的突破点。计算机试验是指在计算机上利用代码来模拟具体的情形,获得试验数据。通过试验寻找一个拟模型来代替现实中的复杂模型是近年来计算机试验的一个热点。计算机试验与实体试验最大的区别是计算机试验的输出结果没有随机性,相同的输入条件下,其输出结果一致。因此计算机试验的设计和分析需要新的准则和方法。
拉丁超立方体设计是具有一维投影均匀性的设计,它能够很好的适应计算机试验输出结果没有随机性的特点,因而被广泛地应用于计算机试验。自从这种试验在1979年被提出之后,很多学者作了大量的工作改进这种设计：保留它的一维投影均匀性,增加其它准则,找出具有其它性质的拉丁超立方体设计。例如增加因子之间的正交性或者增加二维甚至更高维的投影均匀性。在回归模型下,正交因子效应的估计之间是不相关的；增加高维的投影均匀性,能够减小预测的方差。因此增加正交性和均匀性对于正确的分析试验数据是很有必要的。傅里叶多项式模型由Butler (2001)提出,用于试验数据的分析。在科学试验的初期,我们往往对模型所知不多,甚至一无所知,傅里叶多项式模型由于能够很好地逼近多项式模型也能很好地逼近空间模型,而非常适用于模型未知情形,试验设计初期的因子筛选设计的分析。对于傅里叶多项式模型,经过适当的变换,它可以写成多项式回归模型的形式,因此需要寻找具有以下性质的拉丁超立方体设计：所有的线性效应相互正交；所有的线性效应与二阶效应,即平方效应和双线性效应,相互正交。Butler (2001)将因子设计中的分辨度概念引入了拉丁超立方体设计当中,称满足以上条件的设计具有分辨度Ⅳ。
在计算机试验中,同样要面临试验中既有定性因子又有定量因子的情形。这时整个定量因子对应的试验设计部分,我们希望它是一个拉丁超立方体设计且具有一维以上的投影均匀性或者正交性；同时对于每个定性因子组合,希望它所对应的定量因子的设计部分,仍然具有同样的性质。在计算机试验中,尽管计算机试验花费少,但有时时间成本高；像一些有限元模型,计算机程序运行几个星期的时间才能完成一次试验是很常见的。这时采用不同精度的模型来降低试验成本是一个很好的选择。分片拉丁超立方体设计能够很好的解决这些问题,成为近年来计算机试验的研究热点之一,目前尚有许多问题值得研究。
试验数据一旦得到,随之而来的就是选取正确的分析方法对数据进行分析,从而获得有用的信息。标准的分析方法,像方差分析、回归分析等,已经在实践中得到应用并解决了很多问题。但是对于新出现的设计,这些传统的分析方法难以得到令人满意的分析结果。超饱和设计因其能够利用较少的试验次数研究较多的因子效应,而得到很多学者的关注。在超饱和设计的构造方面,近20年里,学者们提出了很多新的最优理论和构造方法。然而相对于构造方面的快速发展,超饱和设计的数据分析亟需更多的研究。目前针对超饱和设计的数据分析,虽然提出了一些方法,但是没有一种是特别令人信服的。如何分析和解释来自超饱和设计的试验数据仍然是一个棘手的问题,特别是针对多响应的情形。现有的超饱和设计分析方法,研究的是只有一个响应的情形,但在实际生活中,具有多响应的超饱和设计的试验很常见,这方面数据分析的研究还是空白。
下面简要介绍一下本文各章的内容。
第一章为引言。简要介绍一些背景知识及一些相关的概念。
第二章给出了傅里叶多项式模型下具有分辨度Ⅳ的正交拉丁超立方设计的构造方法。本章构造的设计绝大多数与Butler (2001)构造的设计具有不同试验次数,是最新的设计。而且我们证明了具有此种正交性的拉丁超立方体设计最多能安排的因子数是试验次数的一半。这些设计对于计算机试验中的因子筛选和建立傅里叶模型非常有用。
第三章给出了基于正交表构造分片拉丁超立方体设计的方法。首先,提出了一种基于对称正交表构造分片拉丁超立方体设计的方法。该方法所得到的分片拉丁超立方体设计不仅具有分片结构,而且具有非常好的低维投影均匀性。同时,每一小片也是一个拉丁超立方体设计,且具有和整体拉丁超立方体设计同样的低维投影均匀性。进一步,我们提出了基于非对称正交表构造分片拉丁超立方体设计的方法。利用该方法构造的设计与基于对称正交表构造的设计具有相似的性质,但是在低维投影均匀性上,该方法可以在不同因子组合上达到不同的低维投影均匀性。本章的构造方法不仅容易实现,而且不同于已有的方法,得到的设计在行数和因子数上都非常灵活。
第四章给出了一种适用于多响应超饱和设计的两阶段变量选择策略。该方法将多元偏最小二乘回归方法与逐步回归方法结合,对多响应超饱和设计进行数据分析。本章的方法相对于单响应超饱和设计的分析方法,能够充分的利用观测阵中的信息,因而更能有效地筛选活跃效应。
第五章对本文的工作进行了总结和讨论。
Scientific experiment is an important tool to learn about the nature, it has been successfully applied in all areas of human investigation. With the numerous development of scientific technology, the research objects involve more and more factors and the underlying relationships among these factors are becoming more and more complex. The conclusions only based on intuition and experience are far from enough. Such situations lead to the birth of a new scientific curriculum—experimental design. To make a plan of scientific experiment, a series of actions should be included:define the research problem, develop a plan of experiment, design an experiment, implement an experiment and analyze the data. Among the series of actions, the design of experiments and the analysis of data are statistical problems. The art of designing an experiment and the art of analyzing an experiment are closely intertwined. A good experimental design should minimize the number of runs to get as much useful information as possible, that is, the information should be of benefit to the analysis of data. Data collected from different designs based on different criteria need different analysis methods.
Design of experiments has been developed more than80years since R. A. Fisher started the pioneering work at Rothamsed farm in Britain, in1930s. The modern discipline in design of experiments was stimulated by the research problems in agri-culture and biology, like the famous Mendel's pea experiments. The observations in this kind of experiments are subject to randomization, that is, the observations have fluctuations under the identical experiment setting. Experiment implemented in a laboratory, a factory or a farm, called a physical experiment, is always ac- companied with randomization. A variety of techniques, such as randomization, blocking and replication, are proposed to suit physical experiments. Designs, like orthogonal designs, regression designs, block designs and Latin square designs have successful applications in physical experiments. In the latest decades, benefiting from the rapid advancement of computer technology, the computer has become an increasingly popular tool in the research work. In some cases, traditional physical experiment may be very expensive and time consuming to run an experiment. Even more, some physical experiments are prohibitive, for example, if we want to study the damage brought by a hurricane, it is impossible to do a physical experimenta-tion. The computer experiment is a new breakthrough under such conditions. A computer experiment is a number of runs of the code with various inputs to simu-late the system performance. Often, the codes are computationally expensive to run, and a common objective of an experiment is to fit a cheaper predictor of the output to the data, i.e., find a metamodel to approximate the true complex relationship between the output and input variables. Computer experiments are different from physical experiments because the output is deterministic—rerunning the code with the same inputs gives identical experimental output. So computer experiments need new methodologies of design and analysis.
Latin hypercube designs (LHDs) have a favorite one-dimensional uniformity which is preferable to the deterministic outputs of computer experiments, and they have been almost exclusively recommended in computer experiments. Since the LHD was proposed in1979, many scholars have done a plenty of work to improve LHDs. Designs are first restricted to the class of LHDs and then a second criterion is applied to this class. For example, there is an effort to find designs which are orthogonal or have two or higher-dimensional uniformity within the class of LHDs. Orthogonal property is desirable because it can ensure the independence of estimates of effects corresponding to the orthogonal columns when the analysis is built on a regression model. Two or higher-dimensional uniformity can decrease the variation of prediction. The above two properties can both help to correctly analyze the col- lected data. The Fourier-polynomial model was first proposed by Butler (2001) as a good approximation not only to the polynomial regression model but also to the spa-tial model. In the initial stage of a scientific experiment, to screen the active effects from a lager number of underlying factors is important under model uncertainty. So a Fourier-polynomial model should be a good choice because of its balance between the polynomial regression model and the spatial model. A Fourier-polynomial model can be rewritten to have a form of the polynomial regression model. So we need to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear inter-actions. Butler (2001) applied the resolution criterion of a factorial design to LHDs and defined LHDs with the above mentioned properties to be resolution IV.
It is inevitable that we face computer experiments with both qualitative and quantitative factors. In such an experiment, the whole design is restricted to be an LHD with one or higher-dimensional projective uniformity or orthogonality and each part of the design corresponding to one-level combination of the qualitative factors is also hoped to still have the same properties. Though a computer experiment is economical, it can involve a code that is time-consuming to run. In some finite element models, it is usual for a code to run a few weeks to produce a single response. So there is still a need to reduce the cost. Multi-fidelity computer experiments are good choices for time-consuming experiments. Sliced LHDs can be applied to the above two problems and become a hot research topic in recent years. There are still many issues to be resolved.
Once the data is collected, suitable analysis methods should be taken to get useful information. Standard analysis methods, such as the analysis of variance method, the regression analysis, etc., have already been applied in reality and have solved many problems. But for newly appeared designs, these traditional methods cannot produce satisfactory results. Supersaturated designs (SSDs) have attracted many authors'attention for its strong and powerful competition in run-size economy. Many construction methods have been proposed in recent twenty years. However relative to the rapid development of the construction, the analysis of SSDs needs more investigation. Though a few methods for analyzing the data from SSDs have been proposed, none of them seem very convincing. How to analyze and interpret the data from SSDs is a thorny issue, especially the data from SSDs with multiple responses. The forgoing methods concentrate on the situations where only one response is considered, however in practice, SSDs with multiple responses are often encountered. There is a blank in the analysis of SSDs with multiple responses.
An outline of the dissertation is given in the following.
Chapter1is the introduction, including some background knowledge and definitions that will be used in the following chapters.
Chapter2introduces a convenient and flexible algorithm for con-structing orthogonal LHDs which have resolution IV under the Fourier-polynomial model. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new. In this chapter, we prove that a LHD with resolution IV can study factors with no more than half number of the runs. The newly constructed LHDs are very suitable for factor screening as they require very few experimental runs per factor, and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).
Chapter3presents two construction methods for sliced LHDs based on orthogonal arrays. First a new approach to constructing sliced LHDs is pro-vided based on symmetric orthogonal arrays. The resulting sliced LHDs possess a desirable sliced structure and have an attractive low-dimensional uniformity. Mean-while within each slice, it is also a LHD with the same low-dimensional uniformity. Next, new sliced LHDs are constructed via asymmetric orthogonal arrays. The same desirable properties are possessed as those constructed using symmetric or-thogonal arrays, depending on their combinations, the uniformity may be differed. The construction methods are easy to implement, and unlike the existing methods, the resulting designs are very flexible in run sizes and numbers of factors.
Chapter4proposes a two-stage variable selection strategy for SSDs with multiple responses. The strategy uses the multivariate partial least squares regression in conjunction with the stepwise regression procedure to select true active effects in SSDs with multiple responses. Compared with the analysis methods for SSDs with one response, the information lying in the matrix of observations for the responses is made full use in the MPLS stage so the method is more rational.
Chapter5provides some concluding remarks for the whole disserta-tion.
试验设计源于统计学家R. A. Fisher在上世纪30年代英国Rothamsted农场试验站的开创性工作,至今已有80多年历史。最初的试验设计问题都是受农业和生物学中的问题的激励,像著名的孟德尔豌豆试验。这些试验的特点是试验结果受随机误差的影响,同样的条件,其观测结果会因随机误差而产生波动。在试验室、工厂或者农场中进行的实体试验,都不可避免的具有这种随机性。我们称这样的试验为实体试验。针对实体试验,提出了很多的设计准则：随机化、分区组、重复性、正交性等。基于这些准则的设计,像正交设计、回归设计、区组设计和拉丁方设计,在实践中已经得到成功的应用。近几十年来,随着计算机技术的飞速发展,计算机在研究中得到越来越多的应用。传统的实体试验存在花费高、周期长等问题,甚至一些试验根本不可能进行,像研究飓风破会性问题,这时在计算机上进行模拟试验成为一个新的突破点。计算机试验是指在计算机上利用代码来模拟具体的情形,获得试验数据。通过试验寻找一个拟模型来代替现实中的复杂模型是近年来计算机试验的一个热点。计算机试验与实体试验最大的区别是计算机试验的输出结果没有随机性,相同的输入条件下,其输出结果一致。因此计算机试验的设计和分析需要新的准则和方法。
拉丁超立方体设计是具有一维投影均匀性的设计,它能够很好的适应计算机试验输出结果没有随机性的特点,因而被广泛地应用于计算机试验。自从这种试验在1979年被提出之后,很多学者作了大量的工作改进这种设计：保留它的一维投影均匀性,增加其它准则,找出具有其它性质的拉丁超立方体设计。例如增加因子之间的正交性或者增加二维甚至更高维的投影均匀性。在回归模型下,正交因子效应的估计之间是不相关的；增加高维的投影均匀性,能够减小预测的方差。因此增加正交性和均匀性对于正确的分析试验数据是很有必要的。傅里叶多项式模型由Butler (2001)提出,用于试验数据的分析。在科学试验的初期,我们往往对模型所知不多,甚至一无所知,傅里叶多项式模型由于能够很好地逼近多项式模型也能很好地逼近空间模型,而非常适用于模型未知情形,试验设计初期的因子筛选设计的分析。对于傅里叶多项式模型,经过适当的变换,它可以写成多项式回归模型的形式,因此需要寻找具有以下性质的拉丁超立方体设计：所有的线性效应相互正交；所有的线性效应与二阶效应,即平方效应和双线性效应,相互正交。Butler (2001)将因子设计中的分辨度概念引入了拉丁超立方体设计当中,称满足以上条件的设计具有分辨度Ⅳ。
在计算机试验中,同样要面临试验中既有定性因子又有定量因子的情形。这时整个定量因子对应的试验设计部分,我们希望它是一个拉丁超立方体设计且具有一维以上的投影均匀性或者正交性；同时对于每个定性因子组合,希望它所对应的定量因子的设计部分,仍然具有同样的性质。在计算机试验中,尽管计算机试验花费少,但有时时间成本高；像一些有限元模型,计算机程序运行几个星期的时间才能完成一次试验是很常见的。这时采用不同精度的模型来降低试验成本是一个很好的选择。分片拉丁超立方体设计能够很好的解决这些问题,成为近年来计算机试验的研究热点之一,目前尚有许多问题值得研究。
试验数据一旦得到,随之而来的就是选取正确的分析方法对数据进行分析,从而获得有用的信息。标准的分析方法,像方差分析、回归分析等,已经在实践中得到应用并解决了很多问题。但是对于新出现的设计,这些传统的分析方法难以得到令人满意的分析结果。超饱和设计因其能够利用较少的试验次数研究较多的因子效应,而得到很多学者的关注。在超饱和设计的构造方面,近20年里,学者们提出了很多新的最优理论和构造方法。然而相对于构造方面的快速发展,超饱和设计的数据分析亟需更多的研究。目前针对超饱和设计的数据分析,虽然提出了一些方法,但是没有一种是特别令人信服的。如何分析和解释来自超饱和设计的试验数据仍然是一个棘手的问题,特别是针对多响应的情形。现有的超饱和设计分析方法,研究的是只有一个响应的情形,但在实际生活中,具有多响应的超饱和设计的试验很常见,这方面数据分析的研究还是空白。
下面简要介绍一下本文各章的内容。
第一章为引言。简要介绍一些背景知识及一些相关的概念。
第二章给出了傅里叶多项式模型下具有分辨度Ⅳ的正交拉丁超立方设计的构造方法。本章构造的设计绝大多数与Butler (2001)构造的设计具有不同试验次数,是最新的设计。而且我们证明了具有此种正交性的拉丁超立方体设计最多能安排的因子数是试验次数的一半。这些设计对于计算机试验中的因子筛选和建立傅里叶模型非常有用。
第三章给出了基于正交表构造分片拉丁超立方体设计的方法。首先,提出了一种基于对称正交表构造分片拉丁超立方体设计的方法。该方法所得到的分片拉丁超立方体设计不仅具有分片结构,而且具有非常好的低维投影均匀性。同时,每一小片也是一个拉丁超立方体设计,且具有和整体拉丁超立方体设计同样的低维投影均匀性。进一步,我们提出了基于非对称正交表构造分片拉丁超立方体设计的方法。利用该方法构造的设计与基于对称正交表构造的设计具有相似的性质,但是在低维投影均匀性上,该方法可以在不同因子组合上达到不同的低维投影均匀性。本章的构造方法不仅容易实现,而且不同于已有的方法,得到的设计在行数和因子数上都非常灵活。
第四章给出了一种适用于多响应超饱和设计的两阶段变量选择策略。该方法将多元偏最小二乘回归方法与逐步回归方法结合,对多响应超饱和设计进行数据分析。本章的方法相对于单响应超饱和设计的分析方法,能够充分的利用观测阵中的信息,因而更能有效地筛选活跃效应。
第五章对本文的工作进行了总结和讨论。
Scientific experiment is an important tool to learn about the nature, it has been successfully applied in all areas of human investigation. With the numerous development of scientific technology, the research objects involve more and more factors and the underlying relationships among these factors are becoming more and more complex. The conclusions only based on intuition and experience are far from enough. Such situations lead to the birth of a new scientific curriculum—experimental design. To make a plan of scientific experiment, a series of actions should be included:define the research problem, develop a plan of experiment, design an experiment, implement an experiment and analyze the data. Among the series of actions, the design of experiments and the analysis of data are statistical problems. The art of designing an experiment and the art of analyzing an experiment are closely intertwined. A good experimental design should minimize the number of runs to get as much useful information as possible, that is, the information should be of benefit to the analysis of data. Data collected from different designs based on different criteria need different analysis methods.
Design of experiments has been developed more than80years since R. A. Fisher started the pioneering work at Rothamsed farm in Britain, in1930s. The modern discipline in design of experiments was stimulated by the research problems in agri-culture and biology, like the famous Mendel's pea experiments. The observations in this kind of experiments are subject to randomization, that is, the observations have fluctuations under the identical experiment setting. Experiment implemented in a laboratory, a factory or a farm, called a physical experiment, is always ac- companied with randomization. A variety of techniques, such as randomization, blocking and replication, are proposed to suit physical experiments. Designs, like orthogonal designs, regression designs, block designs and Latin square designs have successful applications in physical experiments. In the latest decades, benefiting from the rapid advancement of computer technology, the computer has become an increasingly popular tool in the research work. In some cases, traditional physical experiment may be very expensive and time consuming to run an experiment. Even more, some physical experiments are prohibitive, for example, if we want to study the damage brought by a hurricane, it is impossible to do a physical experimenta-tion. The computer experiment is a new breakthrough under such conditions. A computer experiment is a number of runs of the code with various inputs to simu-late the system performance. Often, the codes are computationally expensive to run, and a common objective of an experiment is to fit a cheaper predictor of the output to the data, i.e., find a metamodel to approximate the true complex relationship between the output and input variables. Computer experiments are different from physical experiments because the output is deterministic—rerunning the code with the same inputs gives identical experimental output. So computer experiments need new methodologies of design and analysis.
Latin hypercube designs (LHDs) have a favorite one-dimensional uniformity which is preferable to the deterministic outputs of computer experiments, and they have been almost exclusively recommended in computer experiments. Since the LHD was proposed in1979, many scholars have done a plenty of work to improve LHDs. Designs are first restricted to the class of LHDs and then a second criterion is applied to this class. For example, there is an effort to find designs which are orthogonal or have two or higher-dimensional uniformity within the class of LHDs. Orthogonal property is desirable because it can ensure the independence of estimates of effects corresponding to the orthogonal columns when the analysis is built on a regression model. Two or higher-dimensional uniformity can decrease the variation of prediction. The above two properties can both help to correctly analyze the col- lected data. The Fourier-polynomial model was first proposed by Butler (2001) as a good approximation not only to the polynomial regression model but also to the spa-tial model. In the initial stage of a scientific experiment, to screen the active effects from a lager number of underlying factors is important under model uncertainty. So a Fourier-polynomial model should be a good choice because of its balance between the polynomial regression model and the spatial model. A Fourier-polynomial model can be rewritten to have a form of the polynomial regression model. So we need to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear inter-actions. Butler (2001) applied the resolution criterion of a factorial design to LHDs and defined LHDs with the above mentioned properties to be resolution IV.
It is inevitable that we face computer experiments with both qualitative and quantitative factors. In such an experiment, the whole design is restricted to be an LHD with one or higher-dimensional projective uniformity or orthogonality and each part of the design corresponding to one-level combination of the qualitative factors is also hoped to still have the same properties. Though a computer experiment is economical, it can involve a code that is time-consuming to run. In some finite element models, it is usual for a code to run a few weeks to produce a single response. So there is still a need to reduce the cost. Multi-fidelity computer experiments are good choices for time-consuming experiments. Sliced LHDs can be applied to the above two problems and become a hot research topic in recent years. There are still many issues to be resolved.
Once the data is collected, suitable analysis methods should be taken to get useful information. Standard analysis methods, such as the analysis of variance method, the regression analysis, etc., have already been applied in reality and have solved many problems. But for newly appeared designs, these traditional methods cannot produce satisfactory results. Supersaturated designs (SSDs) have attracted many authors'attention for its strong and powerful competition in run-size economy. Many construction methods have been proposed in recent twenty years. However relative to the rapid development of the construction, the analysis of SSDs needs more investigation. Though a few methods for analyzing the data from SSDs have been proposed, none of them seem very convincing. How to analyze and interpret the data from SSDs is a thorny issue, especially the data from SSDs with multiple responses. The forgoing methods concentrate on the situations where only one response is considered, however in practice, SSDs with multiple responses are often encountered. There is a blank in the analysis of SSDs with multiple responses.
An outline of the dissertation is given in the following.
Chapter1is the introduction, including some background knowledge and definitions that will be used in the following chapters.
Chapter2introduces a convenient and flexible algorithm for con-structing orthogonal LHDs which have resolution IV under the Fourier-polynomial model. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new. In this chapter, we prove that a LHD with resolution IV can study factors with no more than half number of the runs. The newly constructed LHDs are very suitable for factor screening as they require very few experimental runs per factor, and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).
Chapter3presents two construction methods for sliced LHDs based on orthogonal arrays. First a new approach to constructing sliced LHDs is pro-vided based on symmetric orthogonal arrays. The resulting sliced LHDs possess a desirable sliced structure and have an attractive low-dimensional uniformity. Mean-while within each slice, it is also a LHD with the same low-dimensional uniformity. Next, new sliced LHDs are constructed via asymmetric orthogonal arrays. The same desirable properties are possessed as those constructed using symmetric or-thogonal arrays, depending on their combinations, the uniformity may be differed. The construction methods are easy to implement, and unlike the existing methods, the resulting designs are very flexible in run sizes and numbers of factors.
Chapter4proposes a two-stage variable selection strategy for SSDs with multiple responses. The strategy uses the multivariate partial least squares regression in conjunction with the stepwise regression procedure to select true active effects in SSDs with multiple responses. Compared with the analysis methods for SSDs with one response, the information lying in the matrix of observations for the responses is made full use in the MPLS stage so the method is more rational.
Chapter5provides some concluding remarks for the whole disserta-tion.
引文
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[3]Beattie, S. D., Fong, D. K. H. and Lin, D. K. J. (2002). A two-stage Bayesian model selection strategy for supersaturated designs. Technometrics 44(1),55-63.
[4]Beattie, S. D. and Lin, D. K. J. (2004). Rotated factorial design for computer experiments. J. Chinese Statist. Assoc.42,289-308.
[5]Beattie, S. D. and Lin, D. K. J. (2005). A new class of Latin hypercube for computer experiments. In:Fan, J., Li, G. (Eds.), Contemporary Multivariate Analysis and Experimental Designs in Celebration of Professor Kai-Tai Fang's 65th Birthday. World Scientific, Singapore, pp.206-226.
[6]Bingham, D., Sitter, R. R. and Tang, B. (2009). Orthogonal and nearly orthog-onal designs for computer experiments. Biometrika 96,51-65.
[7]Booth, K. H. V. and Cox, D. R. (1962). Some systematic supersaturated de-signs. Technometrics 4(4),489-495.
[8]Box, G. and Meyer, R. (1986). An analysis for unreplicated fractional factorials. Technometrics 28(1),11-18.
[9]Butler, N. A. (2001). Optimal and orthogonal Latin hypercube designs for com-puter experiments. Biometrika 88,847-857.
[10]Butler, N. A. (2005). Supersaturated Latin hypercube designs. Comm. Statist. Theory Methods 34,417-428.
[11]Butler, N. A. and Denham, M. C. (2000). The peculiar shrinkage properties of partial least squares regression. J. Roy. Statist. Soc. Ser. B 62(3),585-593.
[12]Chipman, H., Hamada, H. and Wu, C. F. J. (1997). A Bayesian variable-selection approach for analyzing designed experiments with complex aliasing. Technometrics 39(4),372-381.
[13]Cioppa, T. M. and Lucas, T. W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49,45-55.
[14]Dam, E. R., Van Husslage, B. G. M. and Hertog, D. den. (2007). Maximin Latin hypercube designs in two dimensions. Operations Research 55,158-169.
[15]Fang, K. T., Li, R. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall, Boca Raton.
[16]Fang, K. T., Lin, D. K. J. and Liu, M. Q. (2003). Optimal mixed-level super-saturated design. Metrika 58(3),279-291.
[17]Garthwaite, P. H. (1994). An interpretation of partial least squares. J. Amer. Statist. Assoc.89(425),122-127.
[18]Georgiou, S. D. (2008). Modelling by supersaturated designs. Comput. Statist. Data Anal.53(2),428-435.
[19]Georgiou, S. D. (2009). Orthogonal Latin hypercube designs from generalized orthogonal designs. J. Statist. Plann. Inference 139,1530-1540.
[20]Geramita, A. V. and Seberry, J. (1979). Orthogonal Designs:Quadratic Forms and Hadamard Matrices. Marcel Dekker, New York-Basel.
[21]Haaland, B. and Qian, P. Z. G. (2010). An approach to constructing nested space-filling designs for multi-fidelity computer experiments. Statist. Sinica 20, 1063-1075.
[22]He, X. and Qian, P. Z. G. (2011). Nested orthogonal array-based Latin hyper-cube designs. Biometrika 98,721-731.
[23]Helland, I. S. (1990). Partial least squares regression and statistical models. Scand. J. Statist.17(2),97-114.
[24]Holcomb D. R., Montgomery, D. C. and Carlyle, W. M. (2003). Analysis of supersaturated designs. J. Qual. Technol.35(1),13-27.
[25]Hoskuldsson, A. (2001). Variable and subset selection in PLS regression. Chemometrics & Intelligent Laboratory Systems 55,23-38.
[26]Jin, R., Chen, W. and Simpson, T. W. (2000). Comparative stud-ies of metamodeling techniques under multiple modeling criteria.8th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Long Beach, CA, paper 4801.
[27]Jones, B. and Nachtsheim, C. J. (2011). A class of three-level designs for defini-tive screening in the presence of second-order effects. J. Qual. Technol.43,1-15.
[28]Joseph, V. R. and Hung, Y. (2008). Orthogonal-maximin Latin hypercube de-signs. Statist. Sinica 18,171-186.
[29]Kennedy, M. C. and O'Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika 87,1-13.
[30]Li, J. and Qian, P. Z. G. (2012). Construction of nested (nearly) orthogonal designs for computer experiments. Statist. Sinica 23,451-466.
[31]Li, P., Zhao, S. L. and Zhang R. C. (2010). A cluster analysis selection strategy for supersaturated designs. Comput. Statist. Data Anal.54,1605-1612.
[32]Li, R. and Lin, D. K. J. (2002). Data analysis of supersaturated designs. Statist. Probab. Lett.59(2),135-144.
[33]Li, R. and Lin, D. K. J. (2003). Analysis methods for supersaturated design: some comparisons. J. Data Sci.1(3),249-260.
[34]Lin, C. D., Bingham, D., Sitter, R. R. and Tang, B. (2010). A new and flexible method for constructing designs for computer experiments. Ann. Statist.38, 1460-1477.
[35]Lin, C. D., Mukerjee, R. and Tang, B. (2009). Construction of orthogonal and nearly orthogonal Latin hypercubes. Biometrika 96,243-247.
[36]Lin, D. K. J. (1993). A new class of supersaturated designs. Technometrics 35(1),28-31.
[37]Liu, Y. and Liu, M. Q. (2011). Construction of optimal supersaturated design with large number of levels. J. Statist. Plann. Inference 141,2035-2043.
[38]Liu, Y. and Liu, M. Q. (2012). Construction of equidistant and weak equidistant supersaturated designs. Metrika 75(1),33-53.
[39]Lu, X. and Wu, X. (2004). A strategy of searching active factors in supersatu-rated screening experiments. J. Qual. Technol.36(4),392-399.
[40]MaKay, M. D., Beckman, R. J. and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21,239-245.
[41]Marley, C. J. and Woods, D. C. (2010). A comparison of design and model selection methods for supersaturated experiments. Comput. Statist. Data Anal. 54(12),3158-3167.
[42]Morris, M. D. and Mitchell, T. J. (1995). Exploratory designs for computational experiments. J. Statist. Plann. Inference 43,381-402.
[43]Owen, A. B. (1992). Orthogonal arrays for computer experiments, integration and visualization. Statist. Sinica 2,439-452.
[44]Owen, A. B. (1994). Controlling correlations in Latin hypercube samples. J. Amer. Statist. Assoc.89,1517-1522.
[45]Pang, F., Liu, M. Q. and Lin, D. K. J. (2009). A construction method for orthogonal Latin hypercube designs with prime power levels. Statist. Sinica 19, 1721-1728.
[46]Park, J. S. (1994). Optimal Latin-hypercube designs for computer experiments. J. Statist. Plann. Inference 39,95-111.
[47]Phoa, F. K. H., Pan, Y. H. and Xu, H. (2009). Analysis of supersaturated designs via the Dantzig selector. J. Statist. Plann. Inference 139,2362-2372.
[48]Qian, P. Z. G. (2009). Nested Latin hypercube designs. Biometrika 96,957-970.
[49]Qian, P. Z. G. (2012). Sliced Latin hypercube designs. J. Amer. Statist. Assoc. 107(497),393-399.
[50]Qian, P. Z. G and Ai, M. Y. (2010). Nested Lattice sampling:a new sampling scheme derived by randomizing nested orthogonal arrays. J. Amer. Statist. As-soc.105,1147-1155.
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