就某些產品或製程而言，其品質特性可藉由反應變數與一個以上解釋變數間的函數關係來描述，這種函數關係稱之為輪廓(profile)。而透過管制圖來觀察此函數關係是否隨著時間改變而發生變化的過程稱為輪廓監控(profile monitoring)。本研究係針對具輪廓內相關之資料建構參數模型，並估計每一時間點的輪廓資料參數，進而監控這些參數是否隨著時間而改變。過去針對輪廓資料的研究中常假設輪廓內觀察值彼此之間互相獨立，但由於製造或資料收集過程等因素導致某些輪廓資料無法滿足此一假設。故本研究先以Tweedie 指數分散隨機過程描述此輪廓內相關(within-profile correlation)，接著在階段Ⅱ利用多變量指數加權移動平均管制圖(multivariate exponentially weighted moving average; MEWMA)建構輪廓管制圖以監控模型主要參數，並藉此判斷製程是否呈穩定狀態。
在利用Tweedie 指數分散過程模型配適線性輪廓資料的模擬分析中，我們考慮不同冪參數、斜率參數及分散參數組合下，比較MEWMA管制圖和Zhang et al. (2014)提出T2與MMR管制圖的表現能力，由統計模擬結果發現本研究所提出之MEWMA管制圖在不同參數組合下的表現能力優於T2與MMR管制圖。最後我們以蘋果生長直徑及吸收度之校準線兩筆資料為例，針對MEWMA管制圖在監控具輪廓內相關之輪廓資料上的適用性做進一步的驗證與說明。

For some manufacturing processes, the quality of a product can be characterized by a functional relationship between the response variable and explanatory variables, which has been referred to as a profile. Recently, profile monitoring is used to maintain the stability of the product quality over time by various industries. In this research, we focus on the single-variate linear profile monitoring. Firstly, the Tweedie exponential dispersion process (Tweedie ED) model is used to describe the correlated relationship among within-profile data. Then, a multivariate exponentially weighted moving average (MEWMA) control chart is constructed to detect the process change in Phase Ⅱ study.

In the simulation studies, different combinations of slope and dispersion parameters are considered in our Tweedie ED model. Moreover, the in-control and out-of-control average run lengths (ARLs) are used as a criteria to evaluation the performance of MEWMA and T^2, MMR control charts. Based on the simulation results, we find that our proposed MEWMA control chart has a better detecting performance than the T^2 and the MMR control charts in Phase Ⅱ study since the Tweedie ED model is more suitable to fit linear profile data if the within-profile data are correlated.

Finally, two numerical examples are given to demonstrate the usefulness of our proposed control charts in practical applications.

1.Crowder, S. V. (1987). A Simple Method for Studying Run–Length Distributions of Exponentially Weighted Moving Average Charts. Technometrics, 29(4), 401-407.
2.Crowder, S. V., & Hamilton, M. D. (1992). An EWMA for Monitoring a Process Standard Deviation. Journal of Quality Technology, 24(1), 12-21.
3.Duan, F., & Wang, G. (2018). Exponential-Dispersion Degradation Process Models With Random Effects and Covariates. IEEE Transactions on Reliability, 67(3), 1128-1142.
4.Dunn, P. K., & Smyth, G. K. (2005). Series Evaluation of Tweedie Exponential Dispersion Model Densities. Statistics and Computing, 15(4), 267-280.
5.Jørgensen, B. (1987). Exponential Dispersion Models. Journal of the Royal Statistical Society: Series B (Methodological), 49(2), 127-145.
6.Jensen, W. A., Birch, J. B., & Woodall, W. H. (2008). Monitoring Correlation Within Linear Profiles Using Mixed Models. Journal of Quality Technology, 40(2), 167-183.
7.Kang, L., & Albin, S. L. (2000). On-Line Monitoring When the Process Yields a Linear Profile. Journal of Quality Technology, 32(4), 418-426.
8.Kim, K., Mahmoud, M. A., & Woodall, W. H. (2003). On the Monitoring of Linear Profiles. Journal of Quality Technology, 35(3), 317-328.
9.Lucas, J. M., & Saccucci, M. S. (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics, 32(1), 1-12.
10.Meeker, W. Q., & Escobar, L. A. (1998). Statistical Methods for Reliability Data. John Wiey & Sons.
11.Mestek, O., Pavlík, J., & Suchánek, M. (1994). Multivariate Control Charts: Control Charts for Calibration Curves. Fresenius' Journal of Analytical Chemistry, 350(6), 344-351.
12.Montgomery, D. C. (2007). Introduction to Statistical Quality Control: John Wiley & Sons.
13.Narvand, A., Soleimani, P., & Raissi, S. (2013). Phase II Monitoring of Auto-Correlated Linear Profiles Using Linear Mixed Model. Journal of Industrial Engineering International, 9(1), 12.
14.Pan, J., Li, C., & Liao, C. (2019). Detecting the Process Change for Nonlinear Profile Data. Quality and Reliability Engineering International.
15.Peng, C. (2015). Inverse Gaussian Processes with Random Effects and Explanatory Variables for Degradation Data. Technometrics, 57(1), 100-111.
16.Qiu, P., Zou, C., & Wang, Z. (2010). Nonparametric Profile Monitoring by Mixed Effects Modeling. Technometrics, 52(3), 265-277.
17.Roberts, S. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239-250.
18.Simon, M. K. (2007). Probability Distributions Involving Gaussian Random Variables: A handbook for Engineers and Scientists: Springer Science & Business Media.
19.Soleimani, P., Noorossana, R., & Amiri, A. (2009). Simple Linear Profiles Monitoring in the Presence of Within Profile Autocorrelation. Computers & Industrial Engineering, 57(3), 1015-1021.
20.Williams, C. K., & Rasmussen, C. E. (1996). Gaussian Processes for Regression. Paper presented at the Advances in Neural Information Processing Systems.
21.Zhang, Y., He, Z., Zhang, C., & Woodall, W. H. (2014). Control Charts for Monitoring Linear Profiles with Within‐Profile Correlation Using Gaussian Process Models. Quality and Reliability Engineering International, 30(4), 487-501.
22.Zhou, S., & Xu, A. (2019). Exponential Dispersion Process for Degradation Analysis. IEEE Transactions on Reliability, 68(2), 398-409.
23.Zou, C., Tsung, F., & Wang, Z. (2007). Monitoring General Linear Profiles Using Multivariate Exponentially Weighted Moving Average Schemes. Technometrics, 49(4), 395-408.