This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmar...This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design. Keywords Centered L2-discrepancy, optimal foldover plan, uniformity, uniformity pattern展开更多
Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering.When some additional resources or information ...Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering.When some additional resources or information become available after the initial design of experiment is carried out,some additional runs and/or factors may be added in the follow-up stage.In this paper,the issue of the uniform row augmented designs and column augmented designs with mixed two-,three-and four-level is investigated.The uniformity of augmented designs is discussed under the wrap-around L_(2)-discrepancy.Some lower bounds of wrap-around L_(2)-discrepancy for the augmented designs are obtained,which can be used to assess uniformity of augmented design.Numerical results show that augmented designs have high efficiency,which have low discrepancy and close to the proposed lower bounds.展开更多
The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the c...The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient Mgorithms for searching the optimal four-level foldover plans.展开更多
基金Supported by SRFDP(Grant No.20090144110002)National Natural Science Foundation of China(Grant No.10671080)+3 种基金Key Project of Chinese Ministry of Education(Grant No.105119)NCET(Grant No.06-672)Scientific Research Plan Item of Hunan Provincial Department of Education(Grant No.10C1091)Innovation Program and Independent Research Project Funded by CCNU
文摘This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design. Keywords Centered L2-discrepancy, optimal foldover plan, uniformity, uniformity pattern
基金supported by the National Natural Science Foundation of China(Nos.12361053,11961027,12161040)Hunan Provincial Natural Science Foundation of China(No.2023JJ30486)Scientific Research Plan Item of Hunan Provincial Department of Education(No.22A0355)。
文摘Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering.When some additional resources or information become available after the initial design of experiment is carried out,some additional runs and/or factors may be added in the follow-up stage.In this paper,the issue of the uniform row augmented designs and column augmented designs with mixed two-,three-and four-level is investigated.The uniformity of augmented designs is discussed under the wrap-around L_(2)-discrepancy.Some lower bounds of wrap-around L_(2)-discrepancy for the augmented designs are obtained,which can be used to assess uniformity of augmented design.Numerical results show that augmented designs have high efficiency,which have low discrepancy and close to the proposed lower bounds.
基金Supported by National Natural Science Foundation of China(Grant Nos.11271147,11471135 and 11471136)
文摘The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient Mgorithms for searching the optimal four-level foldover plans.
