Advertisement

Nonparametric Control Charts Based on Data Depth for Location Parameter

  • M. S. BaraleEmail author
  • D. T. Shirke
Original Article
  • 27 Downloads

Abstract

Traditional control charts like Hotelling \(T^2\) are based on the assumption of multivariate normality and also inapplicable to high-dimensional data. A notion of data depth has been used to measure centrality of a given point in a given data cloud. The data depth inferences do not require multivariate normality and any constraint on the dimension of the data. Liu (J Am Stat Assoc 90(432):1380–1387, 1995) provided control charts for a multivariate processes based on data depth, and the performance of the chart is not reported. There exist few tests for the location parameter of multivariate distribution based on data depth. Using these tests, we proposed nonparametric control charts to detect a shift in the location parameter of the multivariate process. We investigate the performance of the proposed control charts using the average run-length measure for various distributions. Also, the control chart procedure is illustrated by using wine quality data.

Keywords

Multivariate processes Depth functions DD plot Bootstrapping 

Notes

Acknowledgements

Both the authors would like to acknowledge the financial support received from University Grants Commission under Major Research Project (F. No. 43-542/2014 (SR)) to carry out the research work. The second author also would like to thank Department of Science and Technology (DST), Science and Engineering Research Board (SERB), New Delhi, for providing financial support under Extra Mural Research scheme [EMR/2017/167] to carry out the research work.

References

  1. 1.
    Bell RC, Jones-Farmer LA, Billor N (2014) A distribution-free multivariate phase I location control chart for subgrouped data from elliptical distributions. Technometrics 56(4):528–538MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bersimis S, Psarakis SM, Panaretos J (2007) Multivariate statistical process control charts: an overview. Qual Reliab Eng Int 23(5):517–543CrossRefGoogle Scholar
  3. 3.
    Boos DD, Zhang J (2000) Monte Carlo evaluation of resampling-based hypothesis tests. J Am Stat Assoc 95(450):486–492CrossRefGoogle Scholar
  4. 4.
    Bush HM, Chongfuangprinya P, Chen VC, Sukchotrat T, Kim SB (2010) Nonparametric multivariate control charts based on a linkage ranking algorithm. Qual Reliab Eng Int 26(7):663–675CrossRefGoogle Scholar
  5. 5.
    Camci F, Chinnam R, Ellis R (2008) Robust kernel distance multivariate control chart using support vector principles. Int J Prod Res 46(18):5075–5095CrossRefGoogle Scholar
  6. 6.
    Chen N, Zi X, Zou C (2016) A distribution-free multivariate control chart. Technometrics 58(4):448–459MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hotelling H (1947) Multivariate quality control. In: Techniques of statistical analysis. McGraw-Hill, New YorkGoogle Scholar
  8. 8.
    Li J, Liu RY (2004) New nonparametric tests of multivariate locations and scales using data depth. Stat Sci 19:686–696MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18:405–414MathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu RY (1995) Control charts for multivariate processes. J Am Stat Assoc 90(432):1380–1387MathSciNetCrossRefGoogle Scholar
  11. 11.
    Liu RY, Parelius JM, Singh K (1999) Multivariate analysis by data depth: descriptive statistics, graphics and inference (with discussion and a rejoinder by Liu and Singh). Ann Stat 27(3):783–858CrossRefGoogle Scholar
  12. 12.
    Liu RY, Singh K, Teng JH (2004) DDMA-charts: nonparametric multivariate moving average control charts based on data depth. Allg Stat Arch 88(2):235–258MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lowry CA, Montgomery DC (1995) A review of multivariate control charts. IIE Trans 27(6):800–810CrossRefGoogle Scholar
  14. 14.
    Mahalanobis PC (1936) On the generalized distance in statistics. National Institute of Science of India, CalcuttazbMATHGoogle Scholar
  15. 15.
    Marozzi M (2016) Multivariate tests based on interpoint distances with application to magnetic resonance imaging. Stat Methods Med Res 25(6):2593–2610MathSciNetCrossRefGoogle Scholar
  16. 16.
    Oja H (1983) Descriptive statistics for multivariate distributions. Stat Probab Lett 1(6):327–332MathSciNetCrossRefGoogle Scholar
  17. 17.
    Osei-Aning R, Abbasi SA, Riaz M (2017) Bivariate dispersion control charts for monitoring non-normal processes. Qual Reliab Eng Int 33(3):515–529CrossRefGoogle Scholar
  18. 18.
    Phaladiganon P, Kim SB, Chen VC, Baek J-G, Park S-K (2011) Bootstrap-based \(T^2\) multivariate control charts. Commun Stat Simul Comput ® 40(5):645–662CrossRefGoogle Scholar
  19. 19.
    Polansky AM (2005) A general framework for constructing control charts. Qual Reliab Eng Int 21(6):633–653CrossRefGoogle Scholar
  20. 20.
    Shirke D, Khorate S (2017) Power comparison of data depth-based nonparametric tests for testing equality of locations. J Stat Comput Simul 87(8):1489–1497MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tukey JW (1975) Mathematics and the picturing of data. In: Proceedings of the international congress of mathematicians, vol 2. Vancouver, pp 523–531Google Scholar
  22. 22.
    Zuo Y et al (2003) Projection-based depth functions and associated medians. Ann Stat 31(5):1460–1490MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zuo Y, Serfling R (2000) General notions of statistical depth function. Ann Stat 28:461–482MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2019

Authors and Affiliations

  1. 1.Department of StatisticsShivaji UniversityKolhapurIndia

Personalised recommendations