本文針對以線性迴歸模型來描述製程的線性輪廓製程，提出一個Phase I無母數監控方法來取代進行監控時需要隨機誤差服從常態分配假設的有母數監控方法，使得我們所提出的管制圖不但可以更彈性應用在實務監控中，也有較高的監控效率。我們提出的無母數線性輪廓監控方法，不但在隨機誤差不具有常態分配時，能得到較精確的上管制界限，且同時在管制圖偵測到製程中存在失控警訊時，也能夠將潛藏在輪廓製程中發生改變點的位置估計出來。本文是推廣與應用Liu, Zou, and Zhang (2008) 所提出用於監控單一樣本線性輪廓的經驗概似比方法在多個樣本的線性輪廓資料上，結果我們發現所提出的經驗概似比管制圖的真正型 I 誤差機率，在線性輪廓的隨機誤差為非常態分配時，較假設隨機誤差為常態分配的有母數管制圖更接近名義型I誤差機率。我們使用統計模擬的方法來評估所提出管制圖的監控效率，並與存在的有母數管制圖比較，結果顯示在失控狀態為階梯式和漂移式偏移的情況，所提出的經驗概似比管制圖比有母數管制圖有更好的監控效率。最後我們以一個來自鋁電解電容器製程的實際數據來說明所提出的經驗概似比管制圖實務上如何使用。

In this article, based on the linear profile which can be described by a linear regression model, we provide a nonparametric Phase I method to monitor the linear profile process which does not require the random error to follow the normal distribution. The proposed control chart is more flexible in practice and has a higher monitoring efficiency. The nonparametric linear profile monitoring method we proposed can not only get more precise control limits when the random error does not follow the normal distribution, but also estimate the position of the change point in the process at the same time when the control chart detects an out-of-control signal. The proposed empirical likelihood ratio control chart applies and extends the empirical likelihood method which Liu, Zou, and Zhang (2008) proposed to detect a change point in a single linear profile to monitor multiple linear profiles. As a result, the empirical likelihood ratio control chart we suggest has the true type I error probability closer to the nominal type I error probability than the parametric control charts when the random error dose not follow the normal distribution in the linear profiles. We use statistical simulations to evaluate the efficiency of the proposed control chart and compare with the existing parametric control charts. The results show that the proposed control chart has better efficiency than the existing parametric control charts when the out-of-control scenario is the step-shift or drifting. Finally, we use a real example from the aluminum electrolytic capacitor process to illustrate how to implement the proposed empirical likelihood ratio control chart in practice.

目錄

第一章緒論1

1.1 前言1
1.2 簡介2
1.3 研究動機與目的5

第二章 Phase I 的線性輪廓製程監控6
2.1 T^2管制圖8
2.2 Global F Test管制圖10
2.3 經驗概似比管制圖11

第三章管制圖效率的比較17
3.1 階梯式偏移20
3.2 離群值偏移21
3.3 漂移式偏移22
3.4 截距係數偏移23
3.5 改變點的估計24

第四章實例應用24

第五章結論與後續研究29

參考文獻31
附圖34
附表37

[1] Anderson, T. W. and Darling, D. A. (1952). ”Asymptotic Theory of Certain Goodness-of-Fit Criteria Based on Stochastic Process”. Annals of Mathematical Statistics 23, pp. 193-212.
[2] Chang, S. I. and Yadama, S. (2010). ”Statistical Process Control for Monitoring Non-Linear Profiles Using Wavelet Filtering and B-Spline Approximation”. International Journal of Production Research 48, pp. 1049-1068.
[3] Hawkins, D. M. and Deng, Q. (2010). ”A Nonparametric Change Point Control Chart”. Journal of Quality Technology 42(2), pp. 165-173.
[4] Hotelling, H. (1947). ”Multivariate Quality Control”. Techniques of Statistical Analysis, Eisenhart, Hastay, and Wallis (Ed.), McGraw-Hill, New York.
[5] Kang, L. and Albin, S. L. (2000). ”On-Line Monitoring When the Process Yields a Linear Profile”. Journal of Quality Technology 32, pp. 418-426.
[6] Kim, K., Mahmoud, M. A., and Woodall, W. H. (2003). ”On the Monitoring of Linear Profiles”. Journal of Quality Technology 35, pp. 317-328.
[7] Lowry, C. A., Woodall, W. H., Champ, C. W., and Rigdon, S. E. (1992). ”A Multivariate Exponentially Weighted Moving Average Control Chart”. Technometrics 34, pp. 46-53.
[8] Liu, Y., Zou, C., and Zhang, R. (2008). ”Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model”. Communications in Statistics Theory and Methods 37(16), pp. 2551-2563.
[9] Mahmoud, M. A. and Woodall, W. H. (2004). ”Phase I Analysis of Linear Profiles with Calibration Applications”. Technometrics 46, pp. 377-391.
[10] Mahmoud, M. A., Parker, P. A., Woodall, W. H., and Hawkins, D. M. (2007). ”A Change Point Method for Linear Profile Data”. Quality and Reliability Engineering Interational 35(2), pp. 247-268.
[11] Mestek, O., Pavlik, J., and Suchanek, M. (1994). ”Multivariate Control Charts: Control Charts for Calibration Curves”. Fresenius’ Journal of Analytical Chemistry 350(6), pp. 344-351.
[12] Noorossana, R., Vaghefi, A., Amiri, A. H., and Roghanian, H.
(2004). ”Monitoring Quality Characteristics Using Linear Profiles”. In Proceeding of the 2nd International Industrial Engineering Conference, Tehran, Iran.
[13] Owen, A. B. (1988). ”Empirical Likelihood Ratio Confidence Intervals for a Single Functional”. Biometrika 75, pp. 237–249.
[14] Owen, A. B. (1990). ”Empirical Likelihood Confidence Regions”. The Annals of Statistics 18(1), pp. 90-120.
[15] Owen, A. B. (2001). Empirical likelihood. Chapman and Hall, New York.
[16] Page, E. S. (1954). ”Continuous Inspection Schemes”. Biometrics 41, pp. 100-114.
[17] Qiu, P., Zou, C., and Wang, Z. (2010). ”Nonparametric Profile Monitoring by Mixed Effects Modeling”. Technometrics 52, pp. 265-277.
[18] Roberts, S. W. (1959). ”Control Chart Tests Based on Geometric Moving Average”. Technometrics 1, pp. 239-250.
[19] Shewhart, W. A. (1924). ”Some Application of Statistical Methods to the Analysis of Physical and Engineering Data”. Bell System Technical Journal 3, pp. 43-87.
[20] Stover, F. S. and Brill, R. V. (1998). ”Statistical Quality Control Applied to Ion Chromatography Calibrations”. Journal of Chromatography A 804, pp. 37-43.
[21] Williams, J. D., Woodall, W. H., and Birch, J. B. (2007). ”Statistical Monitoring of Nonlinear Product and Process Quality Profiles”. Quality and Reliability Engineering International 23(7), pp. 925-941.
[22] Williams, J. D., Birch, J. B., Woodall, W. H., and Ferry, N. M. (2007). ”Statistical Monitoring of Heteroscedastic Dose-Response Profiles from High- Throughput Screening”. Journal of Agricultural, Biological, and Environmental Statistics 12, pp. 216-235.
[23] Woodall, W. H. and Ncube, M. M. (1985). ”Multivariate CUSUM Quality control Procedures”. Technometrics 27, pp. 285-292.
[24] Zhang, H. and Albin, S. (2009). ”Detecting Outliers in Complex Profiles Using a χ^2 Control Chart Method”. IIE Transactions 41, pp. 335-345.
[25] Zi X., Zou C. and Tsung, F. (2012). ”A Distribution-Free, Robust Method for Monitoring Linear Profiles Using Rank-Based Regression”. IIE Transactions 44(11), pp. 949-963.