Uniform augmented q -level designs
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Uniform augmented q-level designs Zongyi Hu1 · Jiaqi Liu1 · Yi Li1 · Hongyi Li2 Received: 17 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In many practical applications, follow-up experimental designs are commonly used to explore the relationship between inputs and outputs steps by steps. As some additional resources could be available to the experimenter after the first step, some additional runs or factors may be added in the follow-up stage. The issue of uniform augmented q-level designs is investigated in this paper. Using the level permutation technique, the uniformity of augmented q-level designs is discussed under the average mixture discrepancy, and new lower bounds of average mixture discrepancy for augmented designs are obtained, which can be served as the benchmark for searching uniform augmented designs. Numerical examples show that the uniform augmented designs can be constructed with high efficiency. Keywords Augmented design · Level permutation · Average mixture discrepancy · New lower bound · q-level design Mathematics Subject Classification 62K15 · 62K05 · 62K99
1 Introduction In many industrial experiments such as pharmaceutics and manufacturing, a smaller run size of the first stage experiment is preferred due to the limited experimental costs. In design of experiment, the issue on how to reasonably design the experiment scale and make the experiment possessing wide applicability is very important, and it is usually to be considered. Assuming that the analytical results based on the initial design meet experiment goal, then the experiment is complete. Otherwise, more effective information from the experimental system may need to be collected by a follow-up
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Hongyi Li [email protected]
1
College of Finance and Statistics, Hunan University, Changsha 410079, China
2
College of Mathematics and Statistics, Jishou University, Jishou 416000, China
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Z. Hu et al.
design. That is, a reasonable process of experiment needs to be continuously updated stage by stage. A foldover design is a standard follow-up strategy discussed in many literatures, which involves adding a second fraction. Foldover is a technique by reversing signs of one or more factors in the original two-level design and adding new runs to the original design (Ou et al. 2011; Ou and Qin 2019). Since foldover is limited in the sense that it involves twice number of the original design, it will be much more convenient if a partial foldover could be carried out. Semifoldover design is one of the partial foldover designs (Ou et al. 2013). Semifoldover designs are also obtained by reversing signs of one or more factors in the original designs, but adding only half of the new runs to the original designs. Because of the inflexibility of the number of additional runs in foldover or semifoldover designs, augmented design is a popular follow-up strategy in recent researches. Row augmented designs are constructed by adding additional runs in the followup experiments based on the initial experimen
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