IDEAS home Printed from https://ideas.repec.org/a/spr/stpapr/v62y2021i3d10.1007_s00362-019-01143-6.html
   My bibliography  Save this article

Sharp lower bounds of various uniformity criteria for constructing uniform designs

Author

Listed:
  • A. M. Elsawah

    (BNU-HKBU United International College
    Zagazig University)

  • Kai-Tai Fang

    (BNU-HKBU United International College
    The Chinese Academy of Sciences)

  • Ping He

    (BNU-HKBU United International College)

  • Hong Qin

    (Central China Normal University
    Zhongnan University of Economics and Law)

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.

Suggested Citation

  • A. M. Elsawah & Kai-Tai Fang & Ping He & Hong Qin, 2021. "Sharp lower bounds of various uniformity criteria for constructing uniform designs," Statistical Papers, Springer, vol. 62(3), pages 1461-1482, June.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:3:d:10.1007_s00362-019-01143-6
    DOI: 10.1007/s00362-019-01143-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00362-019-01143-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00362-019-01143-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. A. M. Elsawah & Kai-Tai Fang, 2018. "New results on quaternary codes and their Gray map images for constructing uniform designs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(3), pages 307-336, April.
    2. E. Androulakis & K. Drosou & C. Koukouvinos & Y.-D. Zhou, 2016. "Measures of uniformity in experimental designs: A selective overview," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(13), pages 3782-3806, July.
    3. Elsawah, A.M., 2016. "Constructing optimal asymmetric combined designs via Lee discrepancy," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 24-31.
    4. Fred J. Hickernell, 2002. "Uniform designs limit aliasing," Biometrika, Biometrika Trust, vol. 89(4), pages 893-904, December.
    5. Chatterjee, Kashinath & Li, Zhaohai & Qin, Hong, 2012. "Some new lower bounds to centered and wrap-round L2-discrepancies," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1367-1373.
    6. A. M. Elsawah & Hong Qin, 2017. "Optimum mechanism for breaking the confounding effects of mixed-level designs," Computational Statistics, Springer, vol. 32(2), pages 781-802, June.
    7. A. M. Elsawah & Kai-Tai Fang, 2019. "A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs," Journal of Applied Statistics, Taylor & Francis Journals, vol. 46(7), pages 1288-1322, May.
    8. Zhou, Yong-Dao & Ning, Jian-Hui & Song, Xie-Bing, 2008. "Lee discrepancy and its applications in experimental designs," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1933-1942, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Elsawah, A.M. & Qin, Hong, 2015. "Lee discrepancy on symmetric three-level combined designs," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 273-280.
    2. Kang Wang & Zujun Ou & Jiaqi Liu & Hongyi Li, 2021. "Uniformity pattern of q-level factorials under mixture discrepancy," Statistical Papers, Springer, vol. 62(4), pages 1777-1793, August.
    3. Elsawah, A.M. & Qin, Hong, 2015. "A new strategy for optimal foldover two-level designs," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 116-126.
    4. A. M. Elsawah, 2018. "Choice of optimal second stage designs in two-stage experiments," Computational Statistics, Springer, vol. 33(2), pages 933-965, June.
    5. Li, Hongyi & Chatterjee, Kashinath & Li, Bo & Qin, Hong, 2016. "Construction of Sudoku-based uniform designs with mixed levels," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 111-118.
    6. A. M. Elsawah & Kai-Tai Fang, 2018. "New results on quaternary codes and their Gray map images for constructing uniform designs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(3), pages 307-336, April.
    7. Biao Luo & Hongyi Li & Yingying Wei & Zujun Ou, 2022. "Uniform design with prior information of factors under weighted wrap-around $$L_2$$ L 2 -discrepancy," Computational Statistics, Springer, vol. 37(5), pages 2717-2739, November.
    8. Zou, Na & Ren, Ping & Qin, Hong, 2009. "A note on Lee discrepancy," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 496-500, February.
    9. A. M. Elsawah, 2021. "Multiple doubling: a simple effective construction technique for optimal two-level experimental designs," Statistical Papers, Springer, vol. 62(6), pages 2923-2967, December.
    10. E. Androulakis & C. Koukouvinos, 2013. "A new variable selection method for uniform designs," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(12), pages 2564-2578, December.
    11. Yong-Dao Zhou & Hongquan Xu, 2014. "Space-Filling Fractional Factorial Designs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1134-1144, September.
    12. Liuping Hu & Kashinath Chatterjee & Jiaqi Liu & Zujun Ou, 2020. "New lower bound for Lee discrepancy of asymmetrical factorials," Statistical Papers, Springer, vol. 61(4), pages 1763-1772, August.
    13. Siyu Pan & Jie Li & Zujun Ou & Peng Zhu, 2023. "Projection uniformity of nearly balanced designs," Statistical Papers, Springer, vol. 64(5), pages 1699-1720, October.
    14. Elsawah, A.M. & Qin, Hong, 2014. "New lower bound for centered L2-discrepancy of four-level U-type designs," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 65-71.
    15. Li, Peng-Fei & Liu, Min-Qian & Zhang, Run-Chu, 2004. "Some theory and the construction of mixed-level supersaturated designs," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 105-116, August.
    16. Rong-Xian Yue & Kashinath Chatterjee, 2010. "Bayesian U-type design for nonparametric response surface prediction," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(2), pages 219-231, September.
    17. Yong-Dao Zhou & Kai-Tai Fang, 2013. "An efficient method for constructing uniform designs with large size," Computational Statistics, Springer, vol. 28(3), pages 1319-1331, June.
    18. Narayanaswamy Balakrishnan & Hong Qin & Kashinath Chatterjee, 2016. "Generalized projection discrepancy and its applications in experimental designs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(1), pages 19-35, January.
    19. Zou, Na & Gou, Tingxun & Qin, Hong & Chatterjee, Kashinath, 2020. "Generalized foldover method for high-level designs," Statistics & Probability Letters, Elsevier, vol. 164(C).
    20. A. M. Elsawah & Kai-Tai Fang, 2020. "New foundations for designing U-optimal follow-up experiments with flexible levels," Statistical Papers, Springer, vol. 61(2), pages 823-849, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stpapr:v:62:y:2021:i:3:d:10.1007_s00362-019-01143-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.