Skip to main content
Springer Nature Link
Log in

Improving the space-filling behavior of multiple triple designs

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

A space-filling (SF) design gives a good representation of experimental region with even fewer points by selecting its points everywhere in the region with as few gaps as possible. Elsawah (J Comput Appl Math 384:113164, 2021) presented the multiple tripling (MT) technique for constructing a new class of three-level designs, called multiple triple designs (MTDs). The MT technique showed its superiority over the widely used techniques by constructing new large optimal MTDs in an efficient manner using small initial designs (InDs). This paper gives a closer look at the SF behavior of MTDs after all of its factor projections and level permutations (FPs-LPs) that alter their statistical inference abilities. The selection of optimal designs by FPs-LPs of such large MTDs needs millions of trials to test all the possible cases. This paper tries to solve this hard computational problem by building theoretical bridges between the SF behavior of the MTD after all of its FPs-LPs and the behavior of the corresponding InD, which is investigated based on the similarity among its runs, confounding among its factors, and uniformity of its points. This study provides benchmarks to guide the experimenters before using FPs-LPs for improving the SF behavior of MTDs in the full dimension and any low dimension. Moreover, the construction of non-isomorphic MTDs is discussed and a lower bound of the number of non-isomorphic MTDs is given. Finally, numerical studies to support the interesting theoretical findings are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Chen W, Qi ZF, Zhou YD (2015) Constructing uniform designs under mixture discrepancy. Stat Probab Lett 97:76–82

    Article  MathSciNet  Google Scholar 

  • Clark JB, Dean AM (2001) Equivalence of fractional factorial designs. Stat Sin 11:537–547

    MathSciNet  MATH  Google Scholar 

  • Elsawah AM (2016) Constructing optimal asymmetric combined designs via Lee discrepancy. Stat Probab Lett 118:24–31

    Article  MathSciNet  Google Scholar 

  • Elsawah AM (2017) A closer look at de-aliasing effects using an efficient foldover technique. Statistics 51(3):532–557

    Article  MathSciNet  Google Scholar 

  • Elsawah AM (2019a) Designing uniform computer sequential experiments with mixture levels using Lee discrepancy. J Syst Sci Complex 32(2):681–708

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Elsawah AM (2020) Building some bridges among various experimental designs. J Korean Stat Soc 49:55–81

    Article  MathSciNet  Google Scholar 

  • Elsawah AM (2021) An appealing technique for designing optimal large experiments with three-level factors. J Comput Appl Math 384:113164

    Article  MathSciNet  Google Scholar 

  • Elsawah AM (2022) A novel non-heuristic search technique for constructing uniform designs with a mixture of two- and four-level factors: a simple industrial applicable approach. J Korean Stat Soc. https://doi.org/10.1007/s42952-021-00159-9

    Article  Google Scholar 

  • Elsawah AM, Qin H (2015) A new strategy for optimal foldover two-level designs. Stat Probab Lett 103:116–126

    Article  MathSciNet  Google Scholar 

  • Elsawah AM, Tang Y, Fang KT (2019) Constructing optimal projection designs. Statistics 53(6):1357–1385

    Article  MathSciNet  Google Scholar 

  • Elsawah AM, Fang KT, Ke X (2021) New recommended designs for screening either qualitative or quantitative factors. Stat Pap 62(1):267–307

    Article  MathSciNet  Google Scholar 

  • Fang KT (1980) The uniform designs: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3:363–372

    Google Scholar 

  • Fang KT, Hickernell FJ (1995) The uniform design and its applications. Bulletin of the International Statistical Institute, 50th session, book 1. International Statistical Institute, Beijing, pp 333–349

    Google Scholar 

  • Fang KT, Lin DKJ, Winker P, Zhang Y (2000) Uniform design: theory and application. Technometrics 42:237–248

    Article  MathSciNet  Google Scholar 

  • Fang KT, Lu X, Winker P (2003) Lower bounds for centered and wrap-around \({L_2}\)-discrepancies and construction of uniform designs by threshold accepting. J Complex 19:692–711

    Article  Google Scholar 

  • Fang KT, Li RZ, Sudjianto A (2006) Design and modeling for computer experiments. Chapman and Hall/CRC, New York

    MATH  Google Scholar 

  • Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322

    Article  MathSciNet  Google Scholar 

  • Hickernell FJ (1998b) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets, vol 138. Lecture notes in statistics. Springer, New York, pp 109–166

    Chapter  Google Scholar 

  • Hua LK, Wang Y (1981) Applications of number theory to numerical analysis. Springer, Berlin

    MATH  Google Scholar 

  • Lam C, Tonchev VD (1996) Classification of affine resolvable 2-(27,9,4) designs. J Stat Plan Inference 56:187–202

    Article  MathSciNet  Google Scholar 

  • Li H, Qin H (2021) A new strategy of tripling. J Korean Stat Soc 50:565–579

    Article  MathSciNet  Google Scholar 

  • Liang YZ, Fang KT, Xu QS (2001) Uniform design and its applications in chemistry and chemical engineering. Chemom Intell Lab Syst 58:43–57

    Article  Google Scholar 

  • Lu X, Fang KT, Xu QF, Yin JX (2002) Balance pattern and BP-optimal factorial designs. Technical Report-324, Hong Kong Baptist University

  • Ma CX, Fang KT (2001) A note on generalized aberration in factorial designs. Metrika 53:85–93

    Article  MathSciNet  Google Scholar 

  • Niederreiter H (1992) Random generation and quasi-Monte Carlo methods. In: Applied Mathematics. SIAM CBMS-NSF Regional Conference, Philadelphia

  • Qin H, Fang KT (2004) Discrete discrepancy in factorial designs. Metrika 60:59–72

    Article  MathSciNet  Google Scholar 

  • Tang B, Deng LY (1999) Minimum \(G_2\)-aberration for nonregular fractional factorial designs. Ann Stat 27:1914–1926

    MATH  Google Scholar 

  • Tang Y, Xu H (2013) An effective construction method for multi-level uniform designs. J Stat Plan Inference 143:1583–1589

    Article  MathSciNet  Google Scholar 

  • Tang Y, Xu H, Lin DKJ (2012) Uniform fractional factorial designs. Ann Stat 40:891–907

    MathSciNet  MATH  Google Scholar 

  • Wang Y, Fang KT (1981) A note on uniform distribution and experimental design. Chin Sci Bull 26:485–489

    MathSciNet  MATH  Google Scholar 

  • 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

    Google Scholar 

  • Xu H (2003) Minimum moment aberration for nonregular designs and supersaturated designs. Stat Sin 13:691–708

    MathSciNet  MATH  Google Scholar 

  • Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29:549–560

    MathSciNet  MATH  Google Scholar 

  • Xu G, Zhang J, Tang Y (2014) Level permutation method for constructing uniform designs under the wrap-around \(L_2\)-discrepancy. J Complex 30:46–53

    Article  Google Scholar 

  • Xu QS, Xu YD, Li L, Fang KT (2018) Uniform experimental design in chemometrics. J Chemom 32:1–10

    Google Scholar 

  • Yi SY, Zhou YD (2018) Projection uniformity under mixture discrepancy. Stat Probab Lett 140:96–105

    Article  MathSciNet  Google Scholar 

  • Zhou YD, Ning JH, Song XB (2008) Lee discrepancy and its applications in experimental designs. Stat Probab Lett 78:1933–1942

    Article  MathSciNet  Google Scholar 

  • Zhou YD, Fang KF, Ning J-H (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thank the two referees, the Associate Editor and the Editor-in-Chief for their constructive comments that lead to significant improvement of this paper. The author greatly appreciates the kind support of Prof. Kai-Tai Fang. This work is supported by the UIC Research Grants with No. of (R201810, R201912, R202010 and R0400001-22) at BNU-HKBU United International College, Zhuhai, PR China; the Guangdong Higher Education Upgrading Plan (2021-2025) of “Rushing to the Top, Making Up Shortcomings and Strengthening Special Features”; and the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College with No. of (2022B1212010006).

Author information

Authors and Affiliations

  1. Division of Science and Technology, Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai, 519087, China

    A. M. Elsawah

  2. Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai, 519087, China

    A. M. Elsawah

  3. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt

    A. M. Elsawah

Authors
  1. A. M. Elsawah
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to A. M. Elsawah.

Ethics declarations

Conflict of interest

There is no conflict of interest.

Additional information

Communicated by Antonio José Silva Neto.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elsawah, A.M. Improving the space-filling behavior of multiple triple designs. Comp. Appl. Math. 41, 180 (2022). https://doi.org/10.1007/s40314-022-01874-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01874-7

Keywords

Mathematics Subject Classification