統計管制圖是監控製程中產品品質穩定性的重要工具。隨著現代製程的進步，製程中經常需要監控多個的品質特性，因此多變量統計製程的設計日益重要。為能設計一種穩健且能監測多變量平均向量微量改變的製程管制方法，本文建議混合使用多變量指數加權移動平均(MEWMA)及累和管制圖(MCUSUM)，其中各品質特性之邊際分布為常態分布，但是應用關連結構模型描述品質特性之聯合分布。本文藉由蒙地卡羅模擬比較所提方法與MEWMA及MCUSUM管制圖在品質特性失控時所需的平均連串長度。最後利用一個在計算機製程中的一組真實數據展示所提方法的實際應用。

Statistical control charts are important tools for monitoring the stability of product quality in manufacturing processes. With advancements in modern processes, it is of increasing importance to monitor more than one quality characteristic simultaneously, hence emphasizing the need for designing multivariate statistical process control methods. To develop a robust method for detecting subtle changes of mean vectors in multivariate processes, this paper proposes a hybrid approach using Multivariate Exponentially Weighted Moving Average (MEWMA) and Multivariate Cumulative Sum (MCUSUM) charts. The marginal distributions of quality characteristics are assumed to be normal distributions, while a structural model is employed to describe the interrelations among quality characteristics. Through Monte Carlo simulations, the performance of the proposed method is compared with that of MEWMA and MCUSUM control charts on the average run length required when multivariate quality characteristics are out-of-control. Finally, the practical application of the proposed method is illustrated by a real-world dataset arising from a computer manufacturing process.

Abbas, N., et al. (2013). Mixed exponentially weighted moving average–cumulative sum charts for process monitoring. Quality and Reliability Engineering International, 29,
345-356.
Ahmad, H., et al. (2024). Copula-based multivariate EWMA control charts for monitoring
the mean vector of bivariate processes using a mixture model. Communications in Statistics-Theory and Methods, 53, 4211-4234.
Ajadi, J. O., and Riaz, M. (2017). Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring. Communications in Statistics - Theory and Methods,
46, 6980–6993.
Box, G. E., and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal
Statistical Society Series B: Statistical Methodology, 26, 211-243.
Busababodhin, P. and Amphanthong, P.(2016). Copula modelling for multivariate statistical process control: a review. Communications for Statistical Applications and Methods;
23, 497–515.
Crosier, R. B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics; 30, 291–303.
Genest, C. and MacKay, J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. The American Statistician; 40, 280–283.
Genest, C. and Rivest, L.P. (1993). Statistical inference procedures for bivariate
Archimedean copulas. Journal of the American statistical Association; 88, 1034–1043.
Hotelling, H. (1947). Multivatiate quality control-illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis, 111-184
Kolmogorov, A. (1933). Sulla determinazione empirica di una legge didistribuzione. Giorn Dell’inst Ital Degli Att, 4, 89-91.
Kuvattana, S., et al. (2015). Efficiency of bivariate copulas on the CUSUM chart. In Proceedings of the international multiconference of engineers and computer scientists; 2, 18-20.
Kuvattana, S., et al. (2016). Bivariate copulas on the exponentially weighted moving
average control chart. Songklanakarin Journal of Science and Technology; 38, 569-574.
Lowry, C. A., et al. (1992). A multivariate exponentially weighted moving average control chart. Technometrics; 34, 46–53.
Lucas, J. M. (1982). Combined shewhart-cusum quality control schemes. Journal of Quality Technology; 14, 51–59.
Lucas, J. M. and Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics; 32, 1–12.
Laio, F. (2004). Cramer–von Mises and Anderson-Darling goodness of fit tests for extreme value distributions with unknown parameters. Water Resources Research; 40, W09308.
Montgomery, D. C. (2020). Introduction to Statistical Quality Control. John Wiley & Sons
Nelsen, R. B. (2006). An Introduction to Copulas. Springer.
Nidsunkid, S., et al. (2017). The effects of violations of the multivariate normality assumption in multivariate shewhart and mewma control charts. Quality and Reliability
Engineering International; 33, 2563–2576.
Nidsunkid, S., et al. (2018). The performance of mcusum control charts when the multivariate normality assumption is violated. Thailand Statistician; 16, 140–155.
Page, E. S. (1954). Continuous inspection schemes. Biometrika; 41, 100–115.
Roberts, S. (1959). Control chart tests based on geometric moving averages. Technometrics; 1, 239–250.
Sasiwannapong, S., et al. (2022). The efficiency of constructed bivariate copulas for MEWMA and Hotelling’s T2 control charts. Communications in Statistics-Simulationand Computation, 51, 1837-1851.
Shapiro, S. S., and Wilk, M. B. (1965). An analysis of variance test for normality (completesamples). Biometrika; 52, 591-611.
Shewhart, W. A. (1930). Economic quality control of manufactured product 1. Bell System Technical Journal; 9, 364–389.
Sklar, A. (1973). Random variables, joint distribution functions, and copulas. Kybernetika;9, 449–460.
Smirnov, N. (1948). Table for estimating the goodness of fit of empirical distributions. The annals of mathematical statistics, 19, 279-281.
Sukparungsee, S., et al. (2018). Bivariate copulas on the Hotelling’s T^2 control chart.Communications in Statistics-Simulation and Computation, 47, 413-419.
Tiengket, S., et al.(2020). Construction of Bivariate Copulas on the Hotelling’s T^2 control chart. Thailand Statistician; 18, 1-15.
Wang, F. K. (2006). Quality evaluation of a manufactured product with multiple characteristics. Quality and Reliability Engineering international; 22, 225-236.