均匀性模式及其在试验设计中的应用

Uniformity pattern and its applications in experimental designs

均匀性是一个几何概念,通常采用不同的偏差来度量设计的均匀性.但偏差的度量侧重于总体,即考虑所有投影子设计的均匀性总和,而根据效应排序原则,低维投影均匀性应该比高维投影均匀性更重要,基于这一思想许多学者提出了类似于字长型的均匀性模式的新概念,并用它来度量设计的各阶投影均匀性.该文对均匀性模式的文献进行了综述,并给出均匀性模式及相关的最小低阶投影均匀性准则的统一定义.该文还对最小低阶投影均匀性准则与其他各种设计筛选准则之间的联系及其在补设计中的应用等进行了综述,这些联系表明最小低阶投影均匀性准则与因子设计中的广义最小低阶混杂准则、正交性准则、V准则、最近平衡性准则、设计效率准则、E(s2)准则之间是相互等价的,这些准则彼此之间可以互相支持,因此揭示了均匀设计和因子设计之间的内在关系.此外,该文还对现有文献中均匀性模式的下界进行了综述,并将达到下界的MPU设计案例罗列出来以方便实际工作者使用.

Uniformity is known as a geometrical concept usually measured by various kinds of discrepancies. But discrepancies put particular emphasis on totality, namely consideration on the whole uniformity of all sub designs. According to the principle of effect hierarchy, lower dimensional projection uniformity is more important than higher dimensional ones. So many scholars proposed a new concept of uniformity pattern like word length pattern to measure the projection uniformity on different dimensions. This article presents a thorough review on existing relative literature and gives an unified definition for uniformity pattern and the criterion of minimum projection uniformity based on it. The application of MPU criterion on complementary designs and the connections between it as well as many other kinds of optimality criteria are summarized. These connections demonstrates that the MPU criterion is equivalent with criteria of generalized minimum aberration, orthogonality, V, nearest balance, design effciency, E（s2） and etc. In other words, they are supportive for each other, which indicates the internal re lationship between uniform designs and factorial designs. Moreover, the lower bounds of uniformity pattern in existing literature and some illustrative examples of minimum projection uniform designs are concluded for convenience.