ABSTRACT The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.

中文翻译：

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构建最佳投影设计

摘要 许多现实生活实验的早期阶段涉及大量因素，其中只有少数因素处于活动状态。不幸的是，那些早期阶段的最佳全维设计可能具有糟糕的低维预测，并且实验者在进行实验之前不知道哪些因素变得重要。因此，具有良好投影的设计对于因子筛选是可取的。在这方面，出现了一些重要的问题，例如最佳全维设计是否对低维有良好的投影？实验者如何通过关注全维设计的所有预测来衡量全维设计的优劣？全维设计的最优性与其预测的最优性之间是否存在联系？通过理论论证，本文试图通过研究用于筛选名义或定量因素的最佳（平均）投影设计的构造，为这些有趣的问题提供答案。主要结果表明：基于像差和正交性准则，全维设计是最优的当且仅当它是最优投影设计；全维设计通过像差和正交性是最优的当且仅当它是均匀投影设计；无法通过任何标准保证统一的全维设计是最佳投影设计；投影设计通过像差、正交性和均匀性标准是最佳的，如果通过它们中的任何一个标准都是最佳的；并且饱和正交设计具有相同的平均投影性能。