混合水平设计的构造及应用

Experimental design has played a fundamental role in the statistical curriculum, practice and research. It has been successfully applied in many fields of scientific investigation, such as agriculture, chemical manufacturing, medicine and other high-tech industries.. Mukerjee and Wu (2006) pointed out that except for treatment comparison with one- or two-way layouts, these problems involve the study of the effects of multiple input variables on the experimental response. An input variable is called a factor and an experiment is called a factorial experiment or a factorial design. Each factor must have two or more settings so that the effect of factor setting on the response can be explored. A setting is called a level of the factor. Any combination of levels of factors is known as a treatment combination. A treatment combination of levels is also called a run in industrial experimentation. A full factorial design involves all possible treatment combinations. For obvious economic reasons, the full factorial experiment with large size may not be feasible. A practical solution is to choose a fraction of the full factorial for experimentation，which is called fractional factorial design. A fractional factorial design is called regular if it can be constructed by using the defining relations. A factorial design in which the numbers of levels of the factors are all equal is called symmetrical, otherwise it is called asymmetrical or a mixed-level factorial design.. As fractional factorial designs are utmost useful in practice, how to select the optimal design becomes very important. There exist some good criteria to select designs. Box and Hunter (1961a,b) proposed maximum resolution (MR) criterion. Based on the fact that designs with the same resolution may not be distinguished, Fries and Hunter (1980) introduced minimum aberration (MA) criterion. Now MR and MA criterion are extended to mixed-level designs. MR and MA are based on the world length.pattern (WLP), which reflects the property of mean confounding between effects. If an experimenter has the prior knowledge, he can not select optimal designs based on WLP. To solve these There exists one problem on choosing optimal designs. The limited experimental resources lead to a fixed number of runs we can plan, how to estimate the factors and their effects as more as possible? We focus on the construction problems of two kinds of designs. Under the general minimum lower-order confounding (GMC) criterion （Zhang, Li, Zhao, and Ai, 2008), we discuss the construction of optimal designs of mixed two- and four-level designs, mixed s^n and (s^2)^m-level designs. ?????

混合水平的试验在实际中常常遇到，但现有最优设计的结果多是借助计算机搜索得到，理论上的构造很少涉及。本项目主要是根据导师张润楚教授最新提出的一般最小低阶混杂（GMC）研究混合水平设计的构造，具体由二水平与四水平的混合设计到一般素数及素数次幂混合水平设计。同时与最小低阶混杂（MA），最大估计容量（MEC），纯净效应（CE） 准则下的最优混合水平设计进行比较。具有现实意义的是我们将本项目中构造出来的混合水平设计进行分析并加以应用，解决实际中遇到的一些问题，为实际工作者提供方便，提高试验的效率。

在长期的试验和探索中，人们基于效应等级原则（EHP），即低阶效应比高阶效应重要，同价效应同等重要，提出了各种判断准则，主要有最大分辨度准则（MR），最小低阶混杂准则（MA），纯净效应准则（CE）和最大估计容量准则（MEC），但是这些准则在选择最优设计的时候得到的结果有时候并不一致。近年来，张润楚教授等提出的一般最小低阶混杂准则（简称GMC）通过别名效应结构（简称AENP），完整的给出了因子之间的混杂关系，并实现了大部分二水平因子的构造问题。本项目结合GMC的发展，对混合水平的设计进行了研究，主要做了以下几方面的工作。.第一，提出了GMC混合水平设计的最优准则。第二，解决了二水平和四水平混合设计m=1,n>N/4的构造情况，并将构造的结果与其他准则进行了比较。.第三，对S水平的GMC设计进行了研究。.最后，本项目还扩充了在超饱和设计以及响应曲面设计的研究，得到了很好的结果，对试验设计领域的发展有推进作用。

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