DOI QR코드

DOI QR Code

Copula modelling for multivariate statistical process control: a review

  • Received : 2016.11.16
  • Accepted : 2016.11.18
  • Published : 2016.11.30

Abstract

Modern processes often monitor more than one quality characteristic that are referred to as multivariate statistical process control (MSPC) procedures. The MSPC is the most rapidly developing sector of statistical process control and increases interest in the simultaneous inspection of several related quality characteristics. Most multivariate detection procedures based on a multi-normality assumptions are independent, but there are many processes that assume non-normality and correlation. Many multivariate control charts have a lack of related joint distribution. Copulas are tool to construct multivariate modelling and formalizing the dependence structure between random variables and applied in several fields. From copula literature review, there are a few copula to apply in MSPC that have multivariate control charts, and represent a successful tool to identify an out-of-control process. This paper presents various types of copulas modelling for the multivariate control chart. The performance measures of the control chart are the average run length (ARL) and the average number of observations to signal (ANOS). Furthermore, a Monte Carlo simulation is shown when the observations were from an exponential distribution.

Acknowledgement

Supported by : Mahasarakham University

References

  1. Alves CC, Samohyl RW, and Henning E (2010). Application of multivariate cumulative sum control charts (MCUSUM) for monitoring a machining process. In Proceedings of the 16th International Conference on Industrial Engineering and Operations Management, Sao Carlos, Brazil.
  2. Bersimis S, Panaretos J, and Psarakis S (2005). Multivariate statistical process control charts and the problem of interpretation: a short overview and some applications in industry. In Proceedings of the 7th Hellenic European Conference on Computer Mathematics and its Applications, Athens, Greece.
  3. Bersimis S, Psarakis S, and Panaretos J (2007). Multivariate statistical process control: an overview, Quality and Reliability Engineering International, 23, 517-543. https://doi.org/10.1002/qre.829
  4. Bourke PD (1991). Detecting shift in fraction nonconforming using run-length control charts with inspection, Journal of Quality Technology, 23, 225-238. https://doi.org/10.1080/00224065.1991.11979328
  5. Busaba J, Sukparungsee S, Areepong Y, and Mititelu G (2012). Analysis of average run length for CUSUM procedure with negative exponential data, Chiang Mai Journal of Science, 39, 200-208.
  6. Crosier RB (1988). Multivariate generalizations of cumulative sum quality-control schemes, Technometrics, 30, 291-303. https://doi.org/10.1080/00401706.1988.10488402
  7. Dokouhaki P and Noorossana R (2013). A copula Markov CUSUM chart for monitoring the bivariate auto-correlated binary observation, Quality and Reliability Engineering International, 29, 911-919. https://doi.org/10.1002/qre.1450
  8. El-Midany TT, El-Baz MA, and Abd-Elwahed MS (2010). A proposed framework for control charts pattern recognition in multivariate procees using artificial neural networks, Expert Systems with Applications, 37, 1035-1042. https://doi.org/10.1016/j.eswa.2009.05.092
  9. Fatahi AA, Dokouhaki P, and Moghaddam BF (2011). A bivariate control chart based on copula function. In Proceedings of the International Conference on Quality and Reliability (ICQR), Bangkok, Thailand, 292-296.
  10. Fatahi AA, Noorossana R, Dokouhaki P, and Moghaddam BF (2012). Copula-based bivariate ZIP control chart for monitoring rare events, Communications in Statistics - Theory and Methods, 41, 2699-2716. https://doi.org/10.1080/03610926.2011.556296
  11. Genest C and MacKay RJ (1986). The joy of copulas: bivariate distributions with uniform marginals, The American Statistician, 40, 280-283.
  12. Hryniewicz O (2012). On the robustness of the Shewhart control chart to different types of dependencies in data, Frontiers in Statistical Quality Control, 10, 19-33.
  13. Hryniewicz O and Szediw A (2010). Sequential signals on a control chart based on nonparametric statistical tests, Frontiers in Statistical Quality Control, 9, 99-117.
  14. Joe H (1997). Multivariate Models and Dependence Concepts, Chapman & Hall, London.
  15. Joe H (2015). Dependence Modeling with Copulas, CRC Press, Boca Raton, FL.
  16. Khoo BC, Atta MA, and Phua HN (2009). A study on the performances of MEWMA and MCUSUM charts for skewed distributions. In Proceedings of the 10th Islamic Countries Conference on Statistical Science, Cairo, Egypt, 817-822
  17. Kuvattana S, Sukparungsee S, Areepong Y, and Busababodhin P (2015a). Multivariate control charts for copulas modeling. In S. Ao, A. H. Chan, H. Katagiri, and L. Xu (Eds), IAENG Transactions on Engineering Sciences: Special Issue for the International Association of Engineers Conferences 2015 (pp. 371-381), World Scientific Publishing, Singapore.
  18. Kuvattana S, Sukparungsee S, Busababodhin P, and Areepong Y (2015b). Efficiency of bivariate copulas on the CUSUM chart. In Proceedings of the International Multiconference of Engineers and Computer Scientists (IMECS), Hong Kong.
  19. Kuvattana S, Sukparungsee S, Areepong Y, and Busababodhin P (2016). Bivariate copulas on the exponentially weighted moving average control chart, Songklanakarin Journal of Science and Technology Preprint, 38, 569-574.
  20. Larpkiatataworn S (2003). A neural network approach for multi-attribute process control with comparison of two current techniques and guidelines for practical use (Ph.D. Thesis), University of Pittsburgh, PA.
  21. Lowry CA and Montgomery DC (1995). A review of multivariate control charts, IIE Transactions, 27, 800-810. https://doi.org/10.1080/07408179508936797
  22. Lowry CA,Woodall WH, Champ CW, and Rigdon SE (1992). A multivariate exponentially weighted moving average control chart, Technometrics, 34, 46-53. https://doi.org/10.2307/1269551
  23. Lu XS (1998). Control chart for multivariate attribute processes, Journal of Production Research, 36, 3477-3489. https://doi.org/10.1080/002075498192166
  24. Marcucci M (1985). Monitoring multinomial processes, Journal of Quality Technology, 17, 86-91. https://doi.org/10.1080/00224065.1985.11978941
  25. Mohmoud MA and Maravelakis PE (2013). The performance of multivariate CUSUM control charts with estimated parameters, Journal of Statistical Computation and Simulation, 83, 721-738. https://doi.org/10.1080/00949655.2011.633910
  26. Montgomery DC (2013). Statistical Quality Control: A Modern Introduction (7th ed), John Wiley & Sons, Singapore.
  27. Nayland College (2004). American new cars and truck, New Zealand.
  28. Nelsen RB (2016). An Introduction to Copulas (2nd ed), Springer, New York.
  29. Niaki STA and Nasaji SA (2011). A hybrid method of artificial neural networks a nd simulated annealing in monitoring auto-correlated multi-attribute processes, International Journal of Advanced Manufacturing Technology, 56, 777-788. https://doi.org/10.1007/s00170-011-3199-4
  30. Patel HI (1973). Quality control methods for multivariate binomial and Poisson distributions, Technometrics, 15, 103-112. https://doi.org/10.1080/00401706.1973.10489014
  31. Petcharat K, Areepong Y, Sukparungsee S, and Mititelu G. (2014). Exact solution for average run length of CUSUM Charts for MA(1) process, Chiang Mai Journal of Science, 41, 1449-1456.
  32. Qiu P (2008). Distribution-free multivariate process control based on log-linear modelling, IIE Transactions, 40, 664-677. https://doi.org/10.1080/07408170701744843
  33. Rao BV, Disney RL, and Pignatiello JJ (2001). Uniqueness and convergence of solutions to average run length integral equations for cumulative sum and other control charts, IIE Transactions, 33, 463-469.
  34. Runger GC, Keats JB, Montgomery DC, and Scranton RD (1999). Improving the performance of the multivariate exponentially weighted moving average control chart, Quality and reliability of Engineering International, 15, 161-166. https://doi.org/10.1002/(SICI)1099-1638(199905/06)15:3<161::AID-QRE215>3.0.CO;2-V
  35. Shih JH and Louis TA (1995). Inferences on the association parameter in copula models for bivariate survival data, Biometrics, 51, 1384-1399. https://doi.org/10.2307/2533269
  36. Sklar M (1959). Fonctions de repartition a n dimensions et leurs marges, Publications de l'lnstitut de Statistique de l'Universite de Paris, 8, 229-231.
  37. Sukparungsee S, Kuvattana S, Areepong Y, and Busababodhin P (2016). Bivariate copulas on the exponentially weighted moving average control chart, Songklanakarin Journal of Science and Technology, 38, 569-574.
  38. Suriyakart W, Areepong Y, Sukparungsee S, Mititelu G (2012). Analytical Method of Average Run Length for Trend Exponential AR(1) Processes inEWMA Procedure, IAENG International Journal of Applied Mathematics, 42, 250-253.
  39. Trivedi PK and Zimmer DM (2005). Copula modeling: an introduction for practitioners, Foundations and Trends in Econometrics, 1, 1-111. https://doi.org/10.1561/0700000001
  40. Verdier G (2013). Application of copulas to multivariate control charts, Journal of Statistical Planning and Inference, 143, 2151-2159. https://doi.org/10.1016/j.jspi.2013.05.005
  41. Xie M and Goh TN (1992). Some procedures for decision making in controlling high yield processes, Quality and Reliability Engineering International, 8, 355-360. https://doi.org/10.1002/qre.4680080409
  42. Xie Y, Xie M, and Goh TN (2011). Two MEWMA charts for Gumbel's bivariate exponential distribution, Journal of Quality Technology, 43, 50-56. https://doi.org/10.1080/00224065.2011.11917845
  43. Zou C and Tsung F (2011). A multivariate sign EWMA control chart, Technometrics, 53, 84-97. https://doi.org/10.1198/TECH.2010.09095

Cited by

  1. Construction of bivariate asymmetric copulas vol.25, pp.2, 2018, https://doi.org/10.29220/CSAM.2018.25.2.217