管制圖常被運用在不同領域中監控機率，在製造業中，會期望製造出不良品的機率越低越好；而在醫療業中，會期望醫療糾紛或罕見疾病發生的機率越低越好，常用來監控機率的管制圖有p管制圖、累積合格品數管制圖等，而適合監控機率的分配分別有伯努力分配、二項分配、幾何分配。本研究以Kullback and Leibler (1951)所提出的Kullback-Leibler information (KLI) 應用幾何分配來建構管制圖(稱作幾何KLI)，與過往提出的管制圖(如修華特管制圖、累積和管制圖、指數加權移動平均管制圖與一般化概似比管制圖)不同的特點在於，幾何KLI管制圖不需要設置管制圖設計參數並且能夠偵測不同範圍的變異，因此對於現場人員的操作更易於上手。本研究提出的幾何KLI管制圖將用於監控不良率並跟Huang et al. (2013)作比較，當監控不良率上升時，幾何KLI針對較大範圍偏移的偵測優於累積和管制圖、指數加權移動平均管制圖與一般化概似比管制圖，且整體績效為最佳，並且擁有不需要設定管制圖設計參數之優勢。在監控不良率下降時，幾何KLI的績效會比幾何累積和管制圖來的好，但兩者績效皆不如伯努力累積和管制圖。在同時監控不良率的上升或下降時，幾何KLI將監控不良率上升與監控不良率下降兩個管制圖結合時，績效會優於伯努力累積和管制圖，且整體績效為最佳。本研究單獨比較幾何KLI與伯努力KLI的績效，在監控不良率上升時，幾何KLI的整體績效為最佳；在監控不良率下降時，伯努力KLI的整體績效為最佳；而在同時監控不良率的上升或下降時，幾何KLI將監控不良率上升與監控不良率下降兩個管制圖結合的整體績效為最佳。最後，本研究以實例示範如何操作幾何KLI。

As a usual technique of quality control, control charts are considered as means of monitoring a process when all items are classified into one of two categories. The proposed control chart based on Kullback-Leibler information (KLI) and a geometric distribution is used for monitoring the nonconforming proportion p of the manufacturing processes in phase II. The chart features a parameter-free setting and rapid detection. The proposed control chart, known as the geometric KLI, has been compared with CUSUM, EWMA, and GLR by Monte Carlo simulation under various shifts when special causes exist in the manufacturing process for detecting an upward shift in p, a downward shift in p, or both. The results turn out that the geometric KLI has an outstanding performance in detecting upward and double-sided shifts. In addition to comparison with different control charts, the geometric KLI will be further discussed with a Bernoulli distributed-based chart through the same measuring method. In the end, a practical application of the geometric KLI in monitoring the adverse rate in orthopedic surgeries will be demonstrated.

Akaike, H. (1974) A new look at the statistical model identification. IEEE transactions on Automatic Control, 19(6), 716-723.
Alwan, L.C., Ebrahimi, N. and Soofi, E.S. (1998) Information theoretic framework for process control. European Journal of Operational Research, 111(3), 526-542.
Borror, C.M., Montgomery, D.C. and Runger, G.C. (1999) Robustness of the EWMA control chart to non-normality. Journal of Quality Technology, 31(3), 309-316.
Bourke, P.D. (1991) Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection. Journal of Quality Technology, 23(3), 225-238.
Bourke, P.D. (2018) Choosing parameter values for a geometric CUSUM chart for detecting an upward shift in a proportion. Quality Engineering, 30(4), 621-634.
Bourke, P.D. (2019) Detecting a downward shift in a proportion using a geometric CUSUM chart. Quality Engineering, 32(1), 75-90.
Calvin, T.W. (1983) Quality-control techniques for “zero defects”. IEEE Transactions on Components Hybrids and Manufacturing Technology, 6(3), 323-328.
Chang, T. and Gan, F. (2001) Cumulative sum charts for high yield processes. Statistica Sinica, 11(3), 791-805.
Chen, G. (1998) An improved p chart through simple adjustments. Journal of Quality Technology, 30(2), 142-151.
Crowder, S.V. (1987) A simple method for studying run–length distributions of exponentially weighted moving average charts. Technometrics, 29(4), 401-407.
Gan, F. (1993) An optimal design of CUSUM control charts for binomial counts. Journal of Applied Statistics, 20(4), 445-460.
Goh, T. (1987) A control chart for very high yield processes. Quality Assurance, 13(1), 18-22.
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), 368-386.
Hawkins, D.M. and Zamba, K. (2005) Statistical process control for shifts in mean or variance using a changepoint formulation. Technometrics, 47(2), 164-173.
Huang, W., Reynolds Jr, M.R. and Wang, S. (2012) A binomial GLR control chart for monitoring a proportion. Journal of Quality Technology, 44(3), 192-208.
Huang, W., Wang, S. and Reynolds Jr, M.R. (2013) A generalized likelihood ratio chart for monitoring Bernoulli processes. Quality and Reliability Engineering International, 29(5), 665-679.
Hurvich, C.M. and Tsai, C. (1989) Regression and time series model selection in small samples. Biometrika, 76(2), 297-307.
Kullback, S. and Leibler, R.A. (1951) On information and sufficiency. Annals of Mathematical Statistics, 22(1), 79-86.
Kullback, S. (1956) Information theory and statistics. Courier Corporation, North Chelmsford, Massachusetts.
Kupperman, M. (1956) Further applications of information-theory to multivariate-analysis and statistical-inference. Annals of Mathematical Statistics, 27(4), 1184-1184.
Lai, T.L. (1995) Sequential changepoint detection in quality control and dynamical systems. Journal of the Royal Statistical Society: Series B (Methodological), 57(4), 613-644.
Lu, S.L. (2018) Non parametric double generally weighted moving average sign charts based on process proportion. Communications in Statistics-Theory and Methods, 47(11), 2684-2700.
Lucas, J.M. and Crosier, R.B. (2000) Fast initial response for CUSUM quality-control schemes: Give your CUSUM a head start. Technometrics, 42(1), 102-107.
Lucas, J.M. and Saccucci, M.S. (1990) Exponentially weighted moving average control schemes: Properties and enhancements. Technometrics, 32(1), 1-12.
Mabude, K., Malela-Majika, J.C., Castagliola, P. and Shongwe, S.C. (2020) Generally weighted moving average monitoring schemes: Overview and perspectives. Quality and Reliability Engineering International, 10.1002.
McCool, J.I. and Joyner-Motley, T. (1998) Control charts applicable when the fraction nonconforming is small. Journal of Quality Technology, 30(3), 240-247.
Reynolds Jr, M.R. and Stoumbos, Z.G. (1999) A CUSUM chart for monitoring a proportion when inspecting continuously. Journal of Quality Technology, 31(1), 87-108.
Reynolds, M.R., Amin, R.W., Arnold, J.C. and Nachlas, J.A. (1988) Charts with variable sampling intervals. Technometrics, 30(2), 181-192.
Reynolds, M.R. and Stoumbos, Z.G. (2001) Individuals control schemes for monitoring the mean and variance of processes subject to drifts. Stochastic Analysis and Applications, 19(5), 863-892.
Roberts, S.W. (2000) Control chart tests based on geometric moving averages. Technometrics, 42(1), 97-101.
Robinson, P. and Ho, T.Y. (1978) Average run lengths of geometric moving average charts by numerical methods. Technometrics, 20(1), 85-93.
Ryan, T.P. and Schwertman, N.C. (1997) Optimal limits for attributes control charts. Journal of Quality Technology, 29(1), 86-98.
Siegmund, D. and Venkatraman, E. (1995) Using the generalized likelihood ratio statistic for sequential detection of a change-point. Annals of Statistics, 23(1), 255-271.
Spliid, H. (2010) An exponentially weighted moving average control chart for Bernoulli data. Quality and Reliability Engineering International, 26(1), 97-113.
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
Sun, J. and Zhang, G. (2000) Control charts based on the number of consecutive conforming items between two successive nonconforming items for the near zero-nonconformity processes. Total Quality Management, 11(2), 235-250.
Szarka III, J.L. and Woodall, W.H. (2012) On the equivalence of the Bernoulli and geometric CUSUM charts. Journal of Quality Technology, 44(1), 54-62.
Tagaras, G. (1998) A survey of recent developments in the design of adaptive control charts. Journal of Quality Technology, 30(3), 212-231.
Van Dobben de Bruyn, C. (1968) Cumulative sum tests: Theory and practice. Griffin, London, England.
Wang, H. (2009) Comparison of p control charts for low defective rate. Computational Statistics & Data Analysis, 53(12), 4210-4220.
Watakabe, K. and Arizono, I. (1999) The power of the ((x)over-bar, s) control chart based on the log‐likelihood ratio statistic. Naval Research Logistics, 46(8), 928-951.
Willsky, A. and Jones, H. (1976) A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems. IEEE Transactions on Automatic Control, 21(1), 108-112.
Woodall, W.H. (1997) Control charts based on attribute data: Bibliography and review. Journal of Quality Technology, 29(2), 172-183.
Wu, Z., Jiao, J. and Liu, Y. (2008) A binomial CUSUM chart for detecting large shifts in fraction nonconforming. Journal of Applied Statistics, 35(11), 1267-1276.
Xie, M., Goh, T. and Lu, X. (1998) Computer-aided statistical monitoring of automated manufacturing processes. Computers & Industrial Engineering, 35(1-2), 189-192.
Xie, W., Xie, M. and Goh, T.N. (1995) A Shewhart-like charting technique for high yield processes. Quality and Reliability Engineering International, 11(3), 189-196.
Yeh, A.B., Mcgrath, R.N., Sembower, M.A. and Shen, Q. (2008) EWMA control charts for monitoring high-yield processes based on non-transformed observations. International Journal of Production Research, 46(20), 5679-5699.