ABSTRACT

Experiments with both qualitative and quantitative factors occur frequently in practical applications. Many construction methods for this kind of designs, such as marginally coupled designs, were proposed to pursue some good space-filling structures. However, few criteria can be adapted to quantify the space-filling property of designs involving both qualitative and quantitative factors. As the uniformity is an important space-filling property of a design, in this paper, a new uniformity criterion, qualitative-quantitative discrepancy (QQD), is proposed for assessing the uniformity of designs with both types of factors. The closed form and lower bounds of the QQD are presented to calculate the exact QQD values of designs and recognize the uniform designs directly. In addition, a connection between the QQD and the balance pattern is derived, which not only helps to obtain a new lower bound but also provides a statistical justification of the QQD. Several examples show that the proposed criterion is reasonable and useful since it can distinguish distinct designs very well.

抽象的

在实际应用中，经常进行定性和定量因素的实验。为了寻求一些好的空间填充结构，人们提出了许多用于这种设计的构造方法，例如边际耦合设计。但是，很少有标准可以用来量化涉及定性和定量因素的设计的空间填充特性。由于均匀性是设计的重要空间填充特性，因此，本文提出了一种新的均匀性标准，即定性-定量-差异（QQD），用于评估两种类型因素的设计均匀性。给出了QQD的封闭形式和下界，以计算设计的精确QQD值并直接识别统一的设计。此外，还导出了QQD与余额模式之间的联系，这不仅有助于获得新的下界，而且还提供了QQD的统计依据。几个例子表明，提出的标准是合理且有用的，因为它可以很好地区分不同的设计。