Abstract
Several techniques are proposed for designing experiments in scientific and industrial areas in order to gain much effective information using a relatively small number of trials. Uniform design (UD) plays a significant role due to its flexibility, cost-efficiency and robustness when the underlying models are unknown. UD seeks its design points to be uniformly scattered on the experimental domain by minimizing the deviation between the empirical and theoretical uniform distribution, which is an NP hard problem. Several approaches are adopted to reduce the computational complexity of searching for UDs. Finding sharp lower bounds of this deviation (discrepancy) is one of the most powerful and significant approaches. UDs that involve factors with two levels, three levels, four levels or a mixture of these levels are widely used in practice. This paper gives new sharp lower bounds of the most widely used discrepancies, Lee, wrap-around, centered and mixture discrepancies, for these types of designs. Necessary conditions for the existence of the new lower bounds are presented. Many results in recent literature are given as special cases of this study. A critical comparison study between our results and the existing literature is provided. A new effective version of the fast local search heuristic threshold accepting can be implemented using these new lower bounds. Supplementary material for this article is available online.
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References
Androulakis E, Drosou K, Koukouvinos C, Zhou YD (2016) Measures of uniformity in experimental designs: a selective overview. Commun Stat Theory Method 45(13):3782–3806
Bates RA, Buck RJ, Riccomagno E, Wynn HP (1996) Experimental design and observation for large systems. J R Stat Soc Ser B 58:77–94
Chatterjee K, Li Z, Qin H (2012a) Some new lower bounds to centered and wrap-round \(L_2\)-discrepancies. Stat Probab Lett 82(7):1367–73
Chatterjee K, Qin H, Na Zou (2012b) Lee discrepancy on two and three mixed level factorials. Sci China 55(3):663–670
Elsawah AM (2016) Constructing optimal asymmetric combined designs via Lee discrepancy. Stat Probab Lett 118:24–31
Elsawah AM (2017a) A closer look at de-aliasing effects using an efficient foldover technique. Statistics 51(3):532–557
Elsawah AM (2017b) A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs. Aust NZ J Stat 59(1):17–41
Elsawah AM (2019a) Building some bridges among various experimental designs. J Korean Stat Soc. https://doi.org/10.1007/s42952-019-00004-0
Elsawah AM (2019b) Constructing optimal router bit life sequential experimental designs: new results with a case study. Commun Stat Simul Comput 48(3):723–752
Elsawah AM, Qin H (2014) New lower bound for centered \(L_2\)-discrepancy of four-level \(U\)-type designs. Stat Probab Lett 93:65–71
Elsawah AM, Qin H (2015a) A new strategy for optimal foldover two-level designs. Stat Probab Lett 103:116–126
Elsawah AM, Qin H (2015b) Mixture discrepancy on symmetric balanced designs. Stat Probab Lett 104:123–132
Elsawah AM, Fang KT (2017) New foundations for designing U-optimal follow-up experiments with flexible levels. Stat Pap. https://doi.org/10.1007/s00362-017-0963-z
Elsawah AM, Qin H (2017) Optimum mechanism for breaking the confounding effects of mixed-level designs. Comput Stat 32(2):781–802
Elsawah AM, Fang KT (2018) New results on quaternary codes and their Gray map images for constructing uniform designs. Metrika 81(3):307–336
Elsawah AM, Fang KT (2019) A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs. J Appl Stat 46(7):1288–1322
Elsawah AM, Fang KT, Ke X (2019) New recommended designs for screening either qualitative or quantitative factors. Stat Pap. https://doi.org/10.1007/s00362-019-01089-9
Fang KT (1980) The uniform designs: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3:363–372
Fang KT, Hickernell FJ (1995) The uniform design and its applications, Bulletin of the International Statistical Institute, 50th Session, Book 1, pp 333–349. International Statistical Institute, Beijing
Fang KT, Mukerjee R (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87:93–198
Fang KT, Lin DKJ, Winker P, Zhang Y (2000) Uniform design: theory and application. Technometrics 42:237–248
Fang KT, Ma CX, Mukerjee R (2002) Uniformity in fractional factorials. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte Carlo and quasi-Monte Carlo methods in scientific computing. Springer, Berlin
Fang KT, Li R, Sudjianto A (2006a) Design and modeling for computer experiments. CRC Press, New York
Fang KT, Maringer D, Tang Y, Winker P (2006b) Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels. Math Comput 75:859–878
Fang KT, Ke X, Elsawah AM (2017) Construction of uniform designs via an adjusted threshold accepting algorithm. J Complex 43:28–37
Fang KT, Tang Y, Yin J (2005) Lower bounds for the wrap-around \(L_2\)-discrepancy of symmetrical uniform designs. J Complex 21:757–771
Fang KT, Wang Y (1994) Number-theoretic methods in statistics. Chapman and Hall, London
Hu L, Chatterjee K, Liu J, Ou Z (2018) New lower bound for Lee discrepancy of asymmetrical factorials. Stat Pap. https://doi.org/10.1007/s00362-018-0998-9
Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322
Hickernell FJ (1998b) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets. Lecture notes in statistics, vol 138. Springer, New York, pp 109–166
Hickernell FJ, Liu M (2002) Uniform designs limit aliasing. Biometrika 89:893–904
Ke X, Zhang R, Ye HJ (2015) Two- and three-level lower bounds for mixture \(L_2\)-discrepancy and construction of uniform designs by threshold accepting. J Complex 31:741–753
Liang YZ, Fang KT, Xu QS (2001) Uniform design and its applications in chemistry and chemical engineering. Chemom Intell Lab Syst 58:43–57
Phadke MS (1986) Design optimization case studies. AT T Tech J 65:51–68
Wang Y, Fang KT (1981) A not on uniform distribution and experimental design. Chin Sci Bull 26:485–489
Winker P, Fang KT (1997) Optimal U-type designs. In: Niederreiter H, Hellekalek P, Larcher G, Zinterhof P (eds) Monte Carlo and quasi-Monte Carlo methods. Springer, New York, pp 436–488
Yang F, Zhou Y-D, Zhang X-R (2017) Augmented uniform designs. J Stat Plan Inference 182:64–737
Zhang Q, Wang Z, Hu J, Qin H (2015) A new lower bound for wrap-around \(L_2\)-discrepancy on two and three mixed level factorials. Stat Probab Lett 96:133–140
Zhou YD, Ning JH (2008) Lower bounds of the wrap-around \(L_2\)-discrepancy and relationships between MLHD and uniform design with a large size. J Stat Plan Inference 138:2330–2339
Zhou YD, Ning JH, Song XB (2008) Lee discrepancy and its applications in experimental designs. Stat Probab Lett 78:1933–1942
Zhou YD, Fang KF, Ning JH (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301
Acknowledgements
The authors greatly appreciate helpful suggestions of the three reviewers, the associate editor and the Editor-in-Chief Professor Werner G. Müller that significantly improved the paper. This work was partially supported by the UIC Grants (Nos. R201810 and R201912), the Zhuhai Premier Discipline Grant and the National Natural Science Foundation of China No. 11871237.
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Elsawah, A.M., Fang, KT., He, P. et al. Sharp lower bounds of various uniformity criteria for constructing uniform designs. Stat Papers 62, 1461–1482 (2021). https://doi.org/10.1007/s00362-019-01143-6
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DOI: https://doi.org/10.1007/s00362-019-01143-6