現今的工業製程日趨複雜，產品或製程的品質特徵可以用一個反應變數以及一個或多個解釋變數的函數關係來界定，這種函數關係被稱為輪廓製程，並使用管制圖進行輪廓製程監控。本文探討當反應變數為類別有序型態且輪廓資料數很小時，我們使用等比機率模型來描述輪廓製程的品質，藉由偵測模型參數是否發生改變來達到改善製程的能力，並利用 Zou 和 Tsung (2011) 提出的多變量符號指數加權移動平均管制圖來監控輪廓製程。接著討論發生在第一階段管制狀態資料組數很小時，多變量符號會受母體分布影響的問題。最後以一個實例來說明本文提的管制圖實務上如何使用。

In complex industrial process, the quality of a product or a
process is better characterized by a relationship (or profile) between the response variable and one or more explanatory variables. The statistical profile monitoring via control chart has gained traction in statistical process control research in recent years. In this article, we focus on monitoring profiles which can be represented by a proportional odds ratio model where the response variable is both categorical and ordinal. An MSEWMA chart discussed in Zou and Tsung (2011) is used to monitor the proportional odds ratio model, especially under the assumption that the subgroup size is small. The chart performance is evaluated via simulation. We also discuss the implementation of the proposed charts when the number of phase I samples is limited. A real example is given to illustrate how these charts can be implemented in practical applications.

第一章 緒論 1
1.1 前言 1
1.2 管制圖介紹 2
1.3 輪廓製程監控 3
1.4 研究動機與目的 4
第二章 輪廓資料監控 6
2.1 等比機率模型 6
2.2 製程中的參數估計 8
2.3 多變量符號EWMA管制圖 10
2.4 其它監控非線性輪廓製程的管制圖 14
2.4.1 利用管制狀態的費雪信息之MEWMA 管制圖 15
2.4.2 利用觀察的費雪信息之MEWMA 管制圖 16
第三章 統計模擬 17
3.1 管制圖的比較 17
3.2 多變量符號EWMA管制圖受第一階段資料組數的影響 24
第四章 實例分析 26
第五章 結論與未來研究 29
參考文獻 30
附表 34
附圖 62

[1] Aly, A. A., Mahmoud, M. A. and Hamed, R. (2015).“The Performance of the Multivariate Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters.” Quality and Reliability Engineering International, doi:10.1002/qre.1806
[2] Chakraborty, B., Chaudhuri, P. and Oja, H(1998). “Operating Transformation Retransformation on Spatial Median and Angle Test.” Statistica Sinica, 8, pp. 767-784.
[3] Cortez, P., Cerdeira, A., Almeida, F., Matos, T. and Reis, J.(2009). “Modeling wine preferences by data mining from physicochemical properties.” Decision support systems, 47, pp. 547-553. Elsevier.
[4] Gandy, A. and Kvaløy, J. T.(2013). “Guaranteed Conditional Performance of Control Charts via Bootstrap Methods.” Scandinavian Journal of Statistics, 40, pp. 647-668.
[5] Han, D. and Tsung, F. (2006). “A reference-free Cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison.” Journal of the American Statistical Association, 101(473), pp. 368-386.
[6] Hettmansperger, T. P. and Randles, R. H. (2002). “A Practical Affine Equivariant Multivariate Median.” Biometrika, 89, pp. 851-860.
[7] Hotelling, H. (1947). “Multivariate Quality Control.” Techniques of Statistical Analysis, Eisenhart, Hastay, and Wallis(ed.), McGraw-Hill, NewYork.
[8] Huwang, L., Wang, Y. -H. T., Yeh, A. B. and Huang, Y. -H. (2015). “Phase II profile monitoring based on proportional odds models.” Computers & Industrial Engineering, 98, pp. 543-553.
[9] 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.
[10] Li, J., Tsung, F. and Zou, C.(2014). “A simple categorical chart for detecting location shifts with ordinal information.” International Journal of Production Research, 52, pp. 550-562.
[11] 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.
[12] McCullagh, P. and Nelder, J. A.(1994). “Generalized linear models.” 2nded. London: Chapman&Hall.
[13] Mestek, O., Pavlik, J. and Suchanek, M.(1944). “Multivariate Control Charts: Control Charts for Calibration Curves.” Journal of Analytical Chemistry, 350, pp. 344-351.
[14] Noorossana, R., Aminnayeri, M. and Izadbakhsh, H.(2013). “Statistical monitoring of polytomous logistic profiles in phase II.” Scientia Iranica, 20, pp. 958-966.
[15] Noorossana, R., Saghaei, A., Izadbakhsh, H. and Aghababaei, O. (2013).“Monitoring multinomial logit profiles via log-linear models.”International Journal of Industrial Engineering and Production Research, 24, pp. 137-142.
[16] Oja, H.(1999). “Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review. ”Scandinavian Journal of Statistics, 26, pp. 319-343
[17] Oja, H. and Randles, R. (2004). “Multivariate Nonparametric Tests.” Statistical Science, 19, pp. 598-605
[18] Page, E. S.(1954). “Continuous Inspection Schemes.” Biometrics, 41, pp. 100-114.
[19] Randles, R. H. (1989). “A Distribution-Free Multivariate Sign Test Based on Interdirections.” Journal of the American Statistical Association, 84, pp. 1045-1050.
[20] Randles, R. H. (2000). “A simpler, affine invariant, multivariate, distribution-free sign test.” Journal of the American Statistical Association, 95, pp. 1263-1268.
[21] Roberts, S. W.(1959). “Control Chart Tests Based on Geometric Moving Average.” Technometrics, 1, pp. 239-250.
[22] Shewhart, W. A.(1924). “Some Applications of Statistical Methods to the Analysis of Physical and Engineering Data.” Bell System Technical Journal, 3, pp. 43-87.
[23] Tyler, D. E. (1987). “A Distribution-Free M-Estimator of Multivariate Scatter.” The Annals of Statistics, 15, pp. 234-251.
[24] 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, pp. 925-941.
[25] Woodall, W. H. and Ncube, M. M.(1985). “Multivariate CUSUM Quality control Procedures.” Technometrics, 27, pp. 285-292.
[26] Zhang, H. and Albin, S. (2009). “Detecting Outliers in Complex Profiles Using a χ2 Control Chart Method.” IIE Transactions, 41, pp. 335-345.
[27] Zou, C., Tsung, F. and Wang, Z.(2007). “Monitoring General Linear Profiles Using
Multivariate Exponential Weighted Moving Average Schemes.” Technometrics, 49, pp. 395-408.
[28] Zou, C., Tsung, F. and Wang, Z. (2008). “Monitoring Profiles Based on Nonparametric Regression Methods.” Technometrics, 50, pp. 512-526
[29] Zou, C. and Tsung, F. (2011). “A Multivariate Sign EWMA Control Chart.” Technometrics, 53:1, pp. 84-97.