The fundamental problem in the orthogonal design theory is the design isomorphism, which involves two classes of methods in the statistical literature. One is to identify the isomorphic designs by costly computation, another is only to detect the non-isomorphic designs as a feasible alternative. In this paper we explore the design structure to propose the degree of isomorphism, as a novel criterion showing the similarity between orthogonal designs. A column-wise framework is proposed to accommodate different issues of the design isomorphism, including the detection of non-isomorphism, identification of isomorphism and determination of subclasses for symmetric orthogonal designs. Our framework shows surprisingly high efficiency, where the average time of identifying the isomorphism between two designs in selected classes is all down to about one second. By applying the hierarchical clustering on the average linkage, a novel classification is also presented for non-isomorphic orthogonal designs in a combinatorial view.

正交设计理论的基本问题是设计同构，它涉及统计文献中的两类方法。一种是通过昂贵的计算来识别同构设计，另一种是仅检测非同构设计作为可行的替代方案。在本文中，我们探索设计结构以提出同构程度，作为显示正交设计之间相似性的新标准。提出了一个按列的框架来适应设计同构的不同问题，包括非同构的检测、同构的识别和对称正交设计的子类的确定。我们的框架显示出惊人的高效率，其中识别所选类中两个设计之间的同构的平均时间都下降到大约一秒。通过在平均链接上应用层次聚类，还为组合视图中的非同构正交设计提出了一种新的分类。