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On Copula Based Serial Dependence in Statistical Process Control

  • Ozlen Erkal SonmezEmail author
  • Alp Baray
Conference paper
Part of the Lecture Notes in Management and Industrial Engineering book series (LNMIE)

Abstract

Copula is a distribution function on the unit hypercube with uniform margins. The margin is directly related to the stochastic behaviour of one variable, while joint distribution function covers the holistic character of more. In multivariate (and particularly bivariate) analysis, using copulas is an elegant way to solve the missing information problem between joint distribution function and the total of the margins. Hereby, the intention of this paper is twofold. Firstly, the paper intends to emphasize the advantages of copulas in practice. In order to encourage potential researchers to diversify their subject of work with these functions, authors give the essential introductory details for a clear understanding of copulas associated with their basic mathematical and statistical preliminaries. Secondly, the study exemplifies the practical usage of copulas in statistical process control area. In this context, process parameters are estimated in order to calculate the control limits of a typical Shewhart type control chart. Parameter estimation is performed by Maximum Likelihood Estimation (MLE) for the bivariate Clayton copula in univariate AR (1) time series with several different levels of high dependence. Since monitoring autocorrelated data in control charts is known as being one of the main causes of producing tighter control limits than required, false alarm rate may be increased and accordingly, the performance of control charts may be dramatically decreased. This study shows that copulas may alternatively be used for getting the same or little wider acceptable region between upper and lower limits. This recognition of the properness of copulas may help to decrease some of the negative effects of dependent data being monitored on charts for further studies.

Keywords

Autocorrelation Control charts Copulas Dependence Statistical process control (SPC) 

References

  1. Albaraccin, O. Y. E., & Alencar, A. P., Ho, L. L. (2017). CUSUM chart to monitor autocorrelated counts using Negative Binomial GARMA model. Statistical Methods in Medical Research, 1–13.  https://doi.org/10.1177/0962280216686627.MathSciNetCrossRefGoogle Scholar
  2. Box, G., & Narashiman, S. (2010). Rethinking STATISTICS FOR QUALITY CONTROL. Quality Engineering, 22, 60–72.CrossRefGoogle Scholar
  3. Busababodhin, P., & Amphanthong, P. (2016). Copula modelling for multivariate statistical process control: A review. Communications for Statistical Applications and Methods, 23, 497–515.CrossRefGoogle Scholar
  4. Chen, J., Yang, H., & Yao, J. (2018). A new multivariate CUSUM chart using principal components with a revision of Crosier’s chart. Communications in Statistics Simulation and Computation, 47, 464–476.MathSciNetCrossRefGoogle Scholar
  5. Del Castillo, E. (2002). Statistical process adjustment for quality control. Wiley Series in Probability and Statistics.Google Scholar
  6. Dokouhaki, P., & Noorossana, R. (2013). A Copula Markov CUSUM chart for monitoring the bivariate auto-correlated binary observation. Quality and Reliability Engineering International, 29, 911–919.CrossRefGoogle Scholar
  7. Embrechts, P. (2009). Copulas: A personal view. The Journal of Risk and Insurance, 76, 639–650.CrossRefGoogle Scholar
  8. Embrechts, P., & Hofert, M. (2012). Copula theory and applications: Quo Vadis? RiskLab ETH Zurich - Swiss Federal Institute of Technology Zurich.Google Scholar
  9. Emura, T., Long, T.-H., & Sun, L.-H. (2016). R routines for performing estimation and statistical process control under copula based time series models. Communications in Statistics Simulation and Computation, 46, 3067–3087.MathSciNetCrossRefGoogle Scholar
  10. Fatahi, A. A., Dokouhaki, P., & Moghaddam, B. F. (2011). A bivariate control chart based on copula function. In IEEE International Conference on Quality and Reliability (pp. 292–296).Google Scholar
  11. Fatahi, A. A., Noorossana, R., Dokouhaki, P., & Moghaddam, B. F. (2012). Copula-based bivariate ZIP control chart for monitoring rare events. Communications in Statistics—Theory and Methods, 41, 2699–2716.MathSciNetCrossRefGoogle Scholar
  12. Franco, B. C., Castagliola, P., Celano, G., & Costa, A. F. B. (2014). A new sampling strategy to reduce the effect of autocorrelation on a control chart. Journal of Applied Statistics, 41, 1408–1421.MathSciNetCrossRefGoogle Scholar
  13. Huang, X., Xu, N., & Bisgaard, S. (2013). A class of Markov chain models for average run length computations for autocorrelated processes. Communications in Statistics Simulation and Computation, 42, 1495–1513.MathSciNetCrossRefGoogle Scholar
  14. Huang, W., Shu, L., Woodall, W. H., & Leung, K. (2016). CUSUM procedures with probability control limits for monitoring processes with variable sample sizes. IEEE Transactions.  https://doi.org/10.1080/0740817x.2016.1146422.CrossRefGoogle Scholar
  15. Hussein, A., Kasem, A., Nkurunziza, S., & Campostrini, S. (2017). Performance of risk adjusted cumulative sum charts when some assumptions are not met. Communications in Statistics Simulation and Computation, 46, 823–830.MathSciNetCrossRefGoogle Scholar
  16. Hryniewicz, O. (2012). On the robustness of the Shewhart chart to different types of dependencies in data. Frontiers in Statistical Quality Control, 10–10, 19–33.CrossRefGoogle Scholar
  17. Hryniewicz, O., & Szediw, A. (2010). Sequential signals on a control chart based on nonparametric statistical tests. Frontiers in Statistical Quality Control, 9, 99–108.CrossRefGoogle Scholar
  18. Kao, S.-C. (2017). A control chart based on weighted bootstrap with strata. Communications in Statistics Simulation and Computation, 47, 1–80.MathSciNetGoogle Scholar
  19. Knoth, S., & Schmid, W. (2004). Control charts for time series: A review. In Frontiers in statistical quality control 7, Springer.Google Scholar
  20. Kuvattana, S., Sukparungsee, S., Busababodhin, P., & Areepong, Y. (2015a). Performance comparison of bivariate copulas on the CUSUM and EWMA control charts. In Proceedings of World Congress on Engineering and Computer Science WCECS: (Vol. 2). ISSN: 2078-0966.Google Scholar
  21. Kuvattana, S., Sukparungsee, S., Busababodhin, P., & Areepong, Y. (2015b). Efficiency of bivariate copula on CUSUM chart. In Proceedings of the International Multi-Conference of Engineers and Computer Scientists IMECS 2015, Hong Kong, (Vol. 2). ISSN: 2087-0966.Google Scholar
  22. Kuvattana, S., Sukparungsee, S., Areepong, Y., & Busababodhin, P. (2015c). Multivariate control charts for copulas modeling. In IAENG transactions on engineering science (pp. 371–381).Google Scholar
  23. Kuvvattana, S., Sukparungsee, S., Areepong, Y., & Busababodhin, P. (2016). Bivariate copulas on the exponentially weighted moving average control chart. Songkllanakarin Journal of Science and Technology Preprint, 38, 569–574.Google Scholar
  24. Kuvattana, S., & Sukparungse, S. (2017). Comperative the performance of control charts based on copulas (pp. 47–58). Springer.Google Scholar
  25. Lee, D., & Joe, H. (2018). Multivariate extreme value copulas with factor and tree dependence structure. Springer Extremes, 21, 147–176.MathSciNetCrossRefGoogle Scholar
  26. Lu, C. W., & Reynolds, M. R. (2001). CUSUM charts for monitoring an autocorrelated process. Journal of Quality Technology, 33, 316–334.CrossRefGoogle Scholar
  27. Lu, S.-L. (2017). Applying fast initial response features on GWMA control charts for monitoring autocorrelated processes. Communications in Statistics: Theory and Methods, 45, 3344–3356.CrossRefGoogle Scholar
  28. Maleki, M. R., Amiri, A., & Castagliola, P. (2017a) Measurement errors in statistical process monitoring: A literature review. Computers and Industrial Engineering, 103, 316–329.CrossRefGoogle Scholar
  29. Maleki, M. R., Amiri, A., & Taheriyoun, A. R. (2017b). Phase II monitoring of binary profiles in the presence of within-profile autocorrelation based on Markov model. Communications in Statistics Simulation and Computation, 46, 7710–7732.MathSciNetCrossRefGoogle Scholar
  30. Nelsen, R. B. (2006). An introduction to copulas. Springer Series in statistics.Google Scholar
  31. Perrone, E., & Müller, W. G. (2016). Optimal designs for copula models. Statistics, 50, 917–929.MathSciNetCrossRefGoogle Scholar
  32. Prajapati, D. R., & Singh, S. (2016). Autocorrelated process monitoring using simple and effective \(\bar{X}\) chart. International Journal of Technology, 85, 929–939.Google Scholar
  33. Psarakis, S., & Papaleonida, G. E. A. (2007). SPC procedures for monitoring autocorrelated processes. Quality Technology and Quantitative Management, 4, 501–540.MathSciNetCrossRefGoogle Scholar
  34. Shu, L., Appley, D. W., & Tsung, F. (2002). Autocorrelated process monitoring using triggered cuscore chart. Quality and Reliability Engineering International, 18, 411–421.CrossRefGoogle Scholar
  35. Sklar, A. (1973). Random variables, joint distribution functions, and copulas. Kybernetika, 9, 449–460.Google Scholar
  36. Sukparungsee, S., Kuvattana, S., Busababodhin, P., & Areepong, Y. (2018). Bivariate copulas on the Hotelling’s T2 control charts. Communications in Statistics Simulation and Computation, 47, 413–419.MathSciNetCrossRefGoogle Scholar
  37. Triantafyllopoulos, K., & Bersimis, S. (2017). Phase II control charts for autocorrelated processes. Quality Technology and Quantitative Management, 13, 88–108.CrossRefGoogle Scholar
  38. Trivedi, P. K. (2005). Copula modeling: An introduction for practitioners. Foundations and Trends in Econometrics.Google Scholar
  39. Verdier, G. (2013). Application of copulas to multivariate control charts. Journal of Statistical Planning, 143, 2151–2159.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Industrial Engineering Department, Faculty of EngineeringIstanbul University - CerrahpasaIstanbulTurkey

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