Skip to main content
SpringerLink
Log in

Optimal foldover plans of three-level designs with minimum wrap-around L 2-discrepancy

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L 2-discrepancy to assess the optimal foldover plans for three-level designs. For three-level fractional factorials as the original designs, the general foldover plan and combined design under a foldover plan are defined, some theoretical properties of the defined foldover plans are obtained, a tight lower bound of the wrap-around L 2-discrepancy of combined designs under a general foldover plan is also obtained, which can be used as a benchmark for searching optimal foldover plans. For illustration of the usage of our theoretical results, a catalog of optimal foldover plans for uniform initial designs with s three-level factors is tabulated, where 2 ⩽ s ⩽ 11.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ai M Y, Hickernell F J, Lin D K J. Optimal foldover plans for regular s-level fractional factorial designs. Statist Probab Lett, 2008, 78: 896–903

    Article  MATH  MathSciNet  Google Scholar 

  2. Box G E P, Hunter W G, Hunter J S. Statistics for Experiments. New York: John Wiley & Sons, 1978

    Google Scholar 

  3. Fang K T, Lin D K J, Qin H. A note on optimal foldover design. Statist Probab Lett, 2003, 62: 245–250

    Article  MATH  MathSciNet  Google Scholar 

  4. Fang K T, Lu X, Winker P. Lower bounds for centered and wrap-around L 2-discrepancies and construction of uniform designs by threshold accepting. J Complexity, 2003, 19: 692–711

    Article  MATH  MathSciNet  Google Scholar 

  5. Fang K T, Tang Y, Yin J X. Lower bounds for wrap-around L 2-discrepancy and constructions of symmetrical uniform designs. J Complexity, 2005, 21: 757–771

    Article  MATH  MathSciNet  Google Scholar 

  6. Hickernell F J. Lattice rules: How well do they measure up. In: Hellekalek P, Larcher G, eds. Random and Quasi-Random Point Sets. New York: Springer-Verlag, 1998, 109–166

    Chapter  Google Scholar 

  7. Lei Y J, Ou Z J, Qin H. Some properties of foldover of regular (s r)×s n fractional factorial designs (in Chinese). Acta Math Sci Ser A Chin Ed, 2011, 31: 978–982

    MATH  MathSciNet  Google Scholar 

  8. Lei Y J, Ou Z J, Qin H, et al. A note on lower bound of centered L 2-discrepancy on combined designs. Acta Math Sin (Engl Ser), 2012, 28: 793–800

    Article  MathSciNet  Google Scholar 

  9. Lei Y J, Qin H, Zou N. Some lower bounds of centered L 2-discrepancy on foldover designs (in Chinese). Acta Math Sci Ser A Chin Ed, 2010, 30: 1555–1561

    MATH  MathSciNet  Google Scholar 

  10. Li F, Jacroux M. Optimal foldover plans for blocked 2m−k fractional factorial designs. J Statist Plann Inference, 2007, 137: 2439–2452

    Article  MATH  MathSciNet  Google Scholar 

  11. Li H, Mee R W. Better foldover fractions for resolution III 2k-p designs. Technometrics, 2002, 44: 278–283

    Article  MathSciNet  Google Scholar 

  12. Li P F, Liu M Q, Zhang R C. Choice of optimal initial designs in sequential experiments. Metrika, 2005, 61: 127–135

    Article  MATH  MathSciNet  Google Scholar 

  13. Li W, Lin D K J. Optimal foldover plans for two-level fractional factorial designs. Technometrics, 2003, 45: 142–149

    Article  MathSciNet  Google Scholar 

  14. Li W, Lin D K J, Ye K Q. Optimal foldover plans for two-level nonregular orthogonal designs. Technometrics, 2003, 45: 347–351

    Article  MathSciNet  Google Scholar 

  15. Montgomery J D, Runger G C. Foldover of 2k-p resolution IV experimental designs. J Qual Technol, 1996, 28: 446–450

    Google Scholar 

  16. Ou Z J, Chatterjee K, Qin H. Lower bounds of various discrepancies on combined designs. Metrika, 2011, 74: 109–119

    Article  MATH  MathSciNet  Google Scholar 

  17. Qin H, Chatterjee K, Ou Z J. A lower bound for the centered L 2-discrepancy on combined designs under the asymmetric factorials. Statistics, 2013, 47: 992–1002

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang B, McLeod R G, Brewster J F. A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs. J Statist Plann Inference, 2010, 140: 1497–1500

    Article  MATH  MathSciNet  Google Scholar 

  19. Ye K Q, Li W. Some properties of blocked and unblocked foldover of 2k-p designs. Statist Sinica, 2003, 13: 403–408

    MATH  MathSciNet  Google Scholar 

  20. Zhang A J, Fang K T, Li R Z, et al. Majorization framework for balanced lattice designs. Ann Statist, 2005, 33: 2837–2853

    Article  MATH  MathSciNet  Google Scholar 

  21. Zhou Y D, Ning J H. Lower bounds of the wrap-around L 2-discrepancy and relationships between MLHD and uniform design with a large size. J Statist Plann Inference, 2008, 138: 2330–2339

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Jishou University, Jishou, 416000, China

    ZuJun Ou

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Hong Qin

  3. School of Physical Science and Technology, Central China Normal University, Wuhan, 430079, China

    ZuJun Ou & Xu Cai

Authors
  1. ZuJun Ou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xu Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong Qin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ou, Z., Qin, H. & Cai, X. Optimal foldover plans of three-level designs with minimum wrap-around L 2-discrepancy. Sci. China Math. 58, 1537–1548 (2015). https://doi.org/10.1007/s11425-014-4936-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4936-6

Keywords

MSC(2010)

Search

Navigation