定性和定量因素同时存在的设计的均匀性准则
Uniformity criterion for designs with both qualitative and quantitative factors
摘要：
在实际应用中，经常会出现定性和定量因素同时存在的实验。为了追求良好的空间填充结构，人们提出了边缘耦合设计等多种构造方法。然而，很少有标准可以适用于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一个重要的空间填充性质，本文提出了一个新的均匀性准则--定性-定量差异（QQD），用于评价两类因子设计的均匀性。给出了QQD的封闭形式和下界，用于计算设计的精确QQD值和直接识别均匀设计。此外，QQD和平衡模式之间的联系，这不仅有助于获得一个新的下界，但也提供了一个统计的理由QQD。算例表明，该准则能很好地区分不同的设计方案，是合理和实用的。
英文摘要：
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.
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