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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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.
Box, G., & Narashiman, S. (2010). Rethinking STATISTICS FOR QUALITY CONTROL. Quality Engineering, 22, 60–72.
Busababodhin, P., & Amphanthong, P. (2016). Copula modelling for multivariate statistical process control: A review. Communications for Statistical Applications and Methods, 23, 497–515.
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.
Del Castillo, E. (2002). Statistical process adjustment for quality control. Wiley Series in Probability and Statistics.
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.
Embrechts, P. (2009). Copulas: A personal view. The Journal of Risk and Insurance, 76, 639–650.
Embrechts, P., & Hofert, M. (2012). Copula theory and applications: Quo Vadis? RiskLab ETH Zurich - Swiss Federal Institute of Technology Zurich.
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.
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).
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.
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.
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.
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.
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.
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.
Hryniewicz, O., & Szediw, A. (2010). Sequential signals on a control chart based on nonparametric statistical tests. Frontiers in Statistical Quality Control, 9, 99–108.
Kao, S.-C. (2017). A control chart based on weighted bootstrap with strata. Communications in Statistics Simulation and Computation, 47, 1–80.
Knoth, S., & Schmid, W. (2004). Control charts for time series: A review. In Frontiers in statistical quality control 7, Springer.
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.
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.
Kuvattana, S., Sukparungsee, S., Areepong, Y., & Busababodhin, P. (2015c). Multivariate control charts for copulas modeling. In IAENG transactions on engineering science (pp. 371–381).
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.
Kuvattana, S., & Sukparungse, S. (2017). Comperative the performance of control charts based on copulas (pp. 47–58). Springer.
Lee, D., & Joe, H. (2018). Multivariate extreme value copulas with factor and tree dependence structure. Springer Extremes, 21, 147–176.
Lu, C. W., & Reynolds, M. R. (2001). CUSUM charts for monitoring an autocorrelated process. Journal of Quality Technology, 33, 316–334.
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.
Maleki, M. R., Amiri, A., & Castagliola, P. (2017a) Measurement errors in statistical process monitoring: A literature review. Computers and Industrial Engineering, 103, 316–329.
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.
Nelsen, R. B. (2006). An introduction to copulas. Springer Series in statistics.
Perrone, E., & Müller, W. G. (2016). Optimal designs for copula models. Statistics, 50, 917–929.
Prajapati, D. R., & Singh, S. (2016). Autocorrelated process monitoring using simple and effective \(\bar{X}\) chart. International Journal of Technology, 85, 929–939.
Psarakis, S., & Papaleonida, G. E. A. (2007). SPC procedures for monitoring autocorrelated processes. Quality Technology and Quantitative Management, 4, 501–540.
Shu, L., Appley, D. W., & Tsung, F. (2002). Autocorrelated process monitoring using triggered cuscore chart. Quality and Reliability Engineering International, 18, 411–421.
Sklar, A. (1973). Random variables, joint distribution functions, and copulas. Kybernetika, 9, 449–460.
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.
Triantafyllopoulos, K., & Bersimis, S. (2017). Phase II control charts for autocorrelated processes. Quality Technology and Quantitative Management, 13, 88–108.
Trivedi, P. K. (2005). Copula modeling: An introduction for practitioners. Foundations and Trends in Econometrics.
Verdier, G. (2013). Application of copulas to multivariate control charts. Journal of Statistical Planning, 143, 2151–2159.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Erkal Sonmez, O., Baray, A. (2019). On Copula Based Serial Dependence in Statistical Process Control. In: Calisir, F., Cevikcan, E., Camgoz Akdag, H. (eds) Industrial Engineering in the Big Data Era. Lecture Notes in Management and Industrial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-03317-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-03317-0_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-03316-3
Online ISBN: 978-3-030-03317-0
eBook Packages: EngineeringEngineering (R0)