品質是一家企業所注重的指標，其中統計製程管制(SPC)是常用的品質管制方法，當所監控的目標可以以輸入與輸出之函數形式呈現，此關係則稱為剖面(profile)，若以線性迴歸模型描述則稱為線性剖面。過去文獻中的線性剖面監控管制圖多半假設在在每個取樣點都需有多個觀察值用以獨立配適出迴歸模型，但在許多實務應用上不一定是好方法，例如低生產率製程。本論文欲探討製程於Phase II階段，用於每次取樣觀察值個數為一個，監控線性迴歸之截距與斜率以及對應之變異數位移之情況，而本論文所提出之管制圖是以Kullback-Leibler information為基礎來監控線性函數是否發生改變，計算統計量的方式則是利用最新一期回溯考慮累積樣本，最後，本論文以平均監控時間(ATS)為管制圖績效指標，透過蒙地卡羅模擬估計管制圖之績效並與其他管制圖比較，並探討群組差異對於管制圖績效的影響。
本研究結果發現，本研究所提出之管制圖若以每次選取多個觀察值群組的方式建構，其在偵測截距項以及斜率項的小幅度參數位移上有較好的表現，而若以單一個觀察值所建構，則可以改善偵測截距項以及斜率項的大幅度偏移上的不足，且在變異數偏移的偵測下不論偏移大小，表現皆優於群組方式建構之管制圖，若同時考量斜率項以及變異數偏移，除了微小幅度的偏移外，皆顯著優於以群組方式所建構之管制圖。最後本研究提供七個AICc模型供使用者判斷偏移類別，隨著偏移幅度變大AICc判斷的正確率越高，且對於變異數有格外敏感之反應。

Statistical process control (SPC) is a commonly used for quality control. When the monitored target can be presented as a function of input and output, this relationship is called a profile. The linear regression model description is called a linear profile.
The profile control charts in the past literature mostly assume that there are several observations at each sampling point, but it is not necessary in many practical applications, such as low productivity processes. This thesis intends to discuss the process in Phase II. The number of observations used for each sampling point is one. The control chart is used to monitor the intercept and slope of linear regression and the corresponding variance. The control chart proposed in this paper based on Kullback-Leibler information (K-L information). It is used to monitor whether the linear function has changed. Finally, this paper uses the average time to signal (ATS) as the performance metric. Through the simulation to present the performance of the control chart and then compared it with other control charts. Finally, we discussed the impact of group differences on the performance of the control chart.
We found that the control chart proposed in this theses, when group at each sampling point, it has a good performance in detecting a small range of shift sizes on intercept term and slope term, and if it is constructed with an individual observation sampling, it can improve the disadvantage in the detection of the large range of shift sizes of the intercept term and slope term, and under the detection of the variance, regardless of the size of the shift, the performance is better than generalized likelihood ration (GLR) control chart. When considering the shift of slope term and variance at the same time, except for the slight range of shift sizes, they are better than the control chart constructed in the group mode. Finally, we provide seven AICc models for users to determine the shift type. As the shift sizes increases, the accuracy of AICc is higher, and there is a particularly sensitive on variance.

參考文獻

中文文獻:
陳長明，以Kullback-Leibler資訊建構剖面監控管制圖，國立成功大學工業與資訊管理研究所碩士論文，民國一百零八年六月。

英文文獻:
Akaike, H. (1974). A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control, 19, 716-723.
Alwan, L. C., Ebrahimi, N., & Soofi, E. S. (1998). Information theoretic framework for process control. European Journal of Operational Research, 111(3), 526-542.
Guo, J., Deng, J., & Wang, Y. (2019). An Intuitionistic Fuzzy Set Based Hybrid Similarity Model for Recommender System. Expert Systems with Applications, 135, 153-163
Gupta, S., Montgomery, D., & Woodall, W. (2006). Performance evaluation of two methods for online monitoring of linear calibration profiles. International Journal of Production Research, 44(10), 1927-1942.
Hurvich, C. M., & Tsai, C.-L. (1989). Regression and time series model selection in small samples. Biometrika, 76(2), 297-307.
Kanagawa, A., Arizono, I., & Ohta, H. (1997). Design of the (x̄, s) control chart based on Kullback-Leibler Information. Frontiers in Statistical Quality Control. Physica-Verlag, Heidelberg, 183-192.
Kang, L., & Albin, S. L. (2000). On-line monitoring when the process yields a linear profile. Journal of Quality Technology, 32(4), 418-426.
Kim, K., Mahmoud, M. A., & Woodall, W. H. (2003). On the monitoring of linear profiles. Journal of Quality Technology, 35(3), 317-328.
Kubokawa, T., & Tsukuma, H. (2007). Estimation in a linear regression model under the Kullback–Leibler loss and its application to model selection. Journal of Statistical Planning and Inference, 137(7), 2487-2508.
Kullback, S. (1978). Information theory and statistics. New York: Dover Publications.
Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79-86.
Kupperman, M. (1956). Further applications of information-theory to multivariate-analysis and statistical-inference. Annals of Mathematical Statistics, 27(4), 1184-1186.
Li, Z., & Wang, Z. (2010). An exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles. Computers & Industrial Engineering, 59(4), 630-637.
Mahmoud, M. A., Morgan, J., & Woodall, W. H. (2010). The monitoring of simple linear regression profiles with two observations per sample. Journal of Applied Statistics, 37(8), 1249-1263.
Montgomery, D. C. (2009). Statistical quality control (Vol. 7): Wiley New York.
Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1/2), 100-115.
Reynolds Jr, M. R., & Lou, J. (2010). An evaluation of a GLR control chart for monitoring the process mean. Journal of Quality Technology, 42(3), 287-310.
Reynolds Jr, M. R., & Stoumbos, Z. G. (2004a). Control charts and the efficient allocation of sampling resources. Technometrics, 46(2), 200-214.
Reynolds Jr, M. R., & Stoumbos, Z. G. (2004b). Should observations be grouped for effective process monitoring? Journal of Quality Technology, 36(4), 343-366.
Reynolds, M. R., Amin, R. W., Arnold, J. C., & Nachlas, J. A. (1988). Charts with variable sampling intervals. Technometrics, 30(2), 181-192.
Roberts, S. (1959). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239-250.
Saghaei, A., Mehrjoo, M., & Amiri, A. (2009). A CUSUM-based method for monitoring simple linear profiles. The International Journal of Advanced Manufacturing Technology, 45(11-12), 1252.
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379-423.
Shewhart, W. A. (1931). Economic control of quality of manufactured product: ASQ Quality Press.
Sugiura, N. (1978). Further analysts of the data by akaike's information criterion and the finite corrections: Further analysts of the data by akaike's. Communications in Statistics-Theory and Methods, 7(1), 13-26.
Tagaras, G. (1998). A survey of recent developments in the design of adaptive control charts. Journal of Quality Technology, 30(3), 212-231.
Wang, S., & Reynolds Jr, M. R. (2013). A GLR control chart for monitoring the mean vector of a multivariate normal process. Journal of Quality Technology, 45(1), 18-33.
Watakabe, K., & Arizono, I. (1999). The power of the (x̄, s) control chart based on the log‐likelihood ratio statistic. Naval Research Logistics, 46(8), 928-951.
Xu, L., Peng, Y., & Reynolds Jr, M. R. (2015). An individuals generalized likelihood ratio control chart for monitoring linear profiles. Quality and Reliability Engineering International, 31(4), 589-599.
Xu, L., Wang, S., Peng, Y., Morgan, J., Reynolds Jr, M. R., & Woodall, W. H. (2012). The monitoring of linear profiles with a GLR control chart. Journal of Quality Technology, 44(4), 348-362.
Zeroual, A., Harrou, F., Sun, Y., & Messai, N. (2018). Integrating Model-Based Observer and Kullback–Leibler Metric for Estimating and Detecting Road Traffic Congestion. IEEE Sensors Journal, 18(20), 8605-8616.
Zhang, J., Li, Z., & Wang, Z. (2009). Control chart based on likelihood ratio for monitoring linear profiles. Computational Statistics & Data Analysis, 53(4), 1440-1448.
Zou, C., Tsung, F., & Wang, Z. (2007). Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics, 49(4), 395-408.