均匀性模式是研究因析设计低维投影性质的重要方法.本文在混合偏差下针对实际应用中最为广泛的二水平、三水平因析设计从投影的角度讨论了其均匀性模式.对于二水平设计获得了其均匀性模式的下界；利用水平置换的方法研究了三水平设计的均匀性模式并获得了其下界.数值例子表明本文中获得的下界均是可达的.
Abstract
Uniformity pattern is an important method to study the low dimension projection of factorial design. Two-and three-level factorial designs are widely used in applications. In this paper, the uniformity pattern of two-and three-level factorial designs is explored from the viewpoint of projection under the mixture discrepancy. The lower bound of uniformity pattern of two-level factorial designs is obtained. The uniformity pattern of three-level factorial designs is studied based on level permutation method and the lower bound of uniformity pattern is obtained. Numerical examples show that the lower bounds obtained in this paper are tight.
关键词
混合偏差 /
投影 /
均匀性模式 /
下界
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Key words
mixture discrepancy /
projection /
uniformity pattern /
lower bound
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参考文献
[1] 雷轶菊, 欧祖军, 李洪毅. 均匀的三水平扩展设计. 应用数学学报, 2018, 41(5):676-688(Lei Y J, Ou Z J, Li H Y. Uniform three-level extended designs. Acta Mathematicae Applicatae Sinica, 2018, 41(5):676-688)
[2] Zhou Y D, Fang K T, Ning J H. Mixture discrepancy for quasi-random point sets. Journal of Complexity, 2013, 29(3-4):283-301
[3] Chen W, Qi Z F, Zhou Y D. Constructing uniform designs under mixture discrepancy. Statistics and Probability Letters, 2015, 97:76-82
[4] Hickernell F J, Liu M Q. Uniform designs limit aliasing. Biometrika, 2002, 89(4):893-904
[5] Yi S Y, Zhou Y D. Projection uniformity under mixture discrepancy. Statistics and Probability Letters, 2018, 140:96-105
[6] Tang Y, Xu H Q, Lin D K J. Uniform fractional factorial designs. Annals of Statistics, 2012, 40:891-907
[7] Zhou Y D, Xu H. Space-filling fractional factorial designs. Journal of the American Statistical Association, 2014, 109(507):1134-1144
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脚注
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基金
国家自然科学基金（11701213，11961027，11561025），湖南省自然科学基金（2020JJ4497），湖南省教育厅重点项目（18A284，19A403）资助.
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