統計製程管制是維持品質穩定的重要工具，其中管制圖可用於偵測製程是否發生異常，並即時通報進行處理。然而，多數傳統管制圖建立於常態分佈假設下，主要用於監控製程平均數或變異數之變化。隨著工程、氣候、水文與金融等領域對極端事件監控需求提升，實際資料可能具有偏態與厚尾性質；若仍以常態管制圖進行監控，偵測能力可能不足且警報延遲，甚至增加誤報與漏報風險。因此，建立適合的極端值管制圖有其必要性。本研究以極值理論中的Gumbel分佈為基礎，結合Kullback-Leibler Information，建立適用於極端值監控之Gumbel KLI管制圖。Gumbel分佈常用於描述日最高溫度、月最大降雨量及年最大風速等極端事件資料；利用Kullback-Leibler Information衡量監控期間之估計分佈與管制內基準分佈間的資訊差異，並搭配由後往前檢定法，逐步回溯過往資料，以降低異常訊號遭稀釋之可能性，提升對分佈參數位移之偵測能力。在參數估計方面，本研究比較最大概似估計法(MLE)、動差法(MoM)與機率加權動差法(PWM)對管制圖績效之影響，並探討僅位置參數位移，以及位置參數與尺度參數同時位移之情境。除此之外，藉蒙地卡羅模擬建立型一錯誤機率與管制內平均連串長度之對應關係，作為管制界線設定依據。結果顯示，當尺度參數往下偏移時，動差法較為合適；當往上偏移時，機率加權動差法具有較佳偵測能力。最後，本研究以交通部中央氣象署台北測站之高溫資料進行案例分析，以平均警報發出所需樣本數做為比較基準；排除有誤報的發生的可能情況外，Gumbel KLI管制圖的平均警報發出所需樣本數皆小於Gumbel分位數管制圖，且KLI管制圖不僅偵測出氣溫異常的變化，亦能回溯變異可能的起始點。本研究方法在案例中具有較佳之偵測時效與結果解釋性，可作為極端值變化監控之一種可行工具。

Control charts provide a systematic way to judge whether observations remain consistent with an expected operating condition. When an unusual pattern appears, the chart can generate a signal for further investigation or corrective action. Many commonly used charts were designed for data that are approximately normally distributed and are primarily intended to identify changes in a process mean or variance. Such assumptions become less suitable when the observations of interest are rare, highly skewed, or concentrated in the upper tail of a distribution. This situation occurs in areas such as environmental monitoring, hydrology, engineering reliability, and financial risk analysis. If a normal-based chart is used without considering these characteristics, the resulting alarm may occur too late or may not adequately distinguish unusual observations from routine variation. A monitoring framework specifically designed for extreme observations is therefore required.
The monitoring rule proposed in this research combines the Gumbel distribution with Kullback Leibler Information (KLI). The Gumbel distribution is appropriate for maxima-type data, including high daily temperatures, monthly peak rainfall, and annual maximum wind speed. For each monitoring point, a Gumbel distribution is fitted to the observations under examination and compared with the in-control reference model. The difference between the two distributions is represented by the KLI statistic. Rather than relying only on a fixed recent sample, the procedure examines observations backward from the latest point. This backward-looking mechanism allows the analysis to focus on a possible abnormal segment and reduces the influence of earlier observations that may still belong to the in-control process. It also provides information regarding the likely beginning of a distributional change.
Three estimation methods are incorporated into the proposed chart: maximum likelihood estimation (MLE), the method of moments (MoM), and probability-weighted moments (PWM). Their performance is examined under changes in the location parameter and under situations in which the location and scale parameters change simultaneously. Control limits are obtained through Monte Carlo experiments by linking the Type I error probability to the desired in-control average run length ($ARL_0$). The simulation results reveal that the preferred estimator varies according to the direction of the parameter change. In particular, MoM tends to give faster signals when the scale parameter becomes smaller. PWM is relatively more effective when the scale parameter increases, although its performance is affected by the number of observations included in the backward-looking segment.
An empirical illustration is performed using daily high-temperature records from the Taipei Weather Station of the Central Weather Administration. The comparison focuses on the average number of observations needed before an alarm is generated. For settings in which false alarms are not observed, the proposed Gumbel KLI chart identifies the unusual temperature pattern using fewer observations, on average, than the Gumbel quantile control charts. The procedure does more than indicate that a signal has occurred; it also helps identify the approximate point at which the abnormal pattern may have started. These results suggest that the proposed approach can be used as an interpretable monitoring tool for extreme-value data.

王柏文、陳昌維 (2016)。台北連3天高溫逾38度 刷新氣象局記錄。https://news.pts.org.tw/article/330288
中央社 (2020)。台北飆38.9度 破歷年7月紀錄。中央社。https://www.cna.com.tw/news/firstnews/202007135005.aspx
中央氣象局 (2014)。101氣候年報。交通部中央氣象署。https://www.cwa.gov.tw/Data/service/notice/download/publish_20150109111232.pdf
中央氣象局 (2015)。103氣候年報。交通部中央氣象署。https://www.cwa.gov.tw/Data/service/notice/download/publish_20160408155429.pdf
中央氣象署 (2020)。2020年7月與2007、2014年並列台灣史上最熱7月。報氣候－中央氣象署。https://www.facebook.com/climate.taiwan/posts/1429692403907953
余曉涵 (2020)。台北高溫飆至39.7度 設站以來歷史最高。中央社。https://www.cna.com.tw/news/firstnews/202007245010.aspx
Abeysingha, N. S., Kularathna, K. M., Bandara, A. M. K. R., & Ray, R. L. (2023). Assessment of extreme rainfall through statistical process control-I chart. Journal of Earth System Science, 132.
Akaike, H. (1974). A new look at statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716-723.
Anojahatlo, K., & Sabri-Laghaie, K. (2022). Enhancing the detection power of multivariate time between events control charts for Gumbel bivariate exponential distribution. Computers & Industrial Engineering, 171.
Borror, C. M., Montgomery, D. C., & 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-percent inspection. Journal of Quality Technology, 23(3), 225-238.
Celano, G., & Castagliola, P. (2020). On‐line monitoring of extreme values of geometric profiles in finite horizon processes. Quality and Reliability Engineering International, 36(4), 1313-1332.
Chang, Y. C., & Wu, Y. C. (2022). A parameter-free exponential control chart based on Kullback-Leibler information for time-between-events monitoring. Quality and Reliability Engineering International, 38(2), 733–756.
Chen, P., Mukherjee, A., Yang, W., & Zhang, J. (2025). Some new real-time monitoring schemes for Gumbel’s bivariate exponential time between the events. Computers & Industrial Engineering, 200.
Chen, P., Song, Z., Hu, X., & Zhang, J. (2024). Phase II control chart for monitoring Gumbel's bivariate exponential distribution. Computers & Industrial Engineering, 192.
Fisher, R. A., & Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24(2), 180-190.
Gan, F. F. (1993). An optimal design of CUSUM control charts for binomial counts. Journal of Applied Statistics, 20(4), 445-460.
Gumbel, E. J. (1942). Statistical control‐curves for flood‐discharges. Eos, Transactions American Geophysical Union, 23(2), 489-509.
Greenwood, J. A., Landwehr, J. M., Matalas, N. C., & Wallis, J. R. (1979). Probability weighted moments. Water Resources Research, 15(5), 1049-1054.
Han, D., & Tsung, F. G. (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(437), 368-386.
Hawkins, D. M., Qiu, P., & Kang, C. W. (2003). The changepoint model for statistical process control. Journal of Quality Technology, 35(4), 355-366.
Hu, X., Xia, F., Teoh, W. L., & Zhang, J. (2025). An adaptive multivariate EWMA chart for monitoring Gumbel's bivariate exponential distributed data. Ain Shams Engineering Journal, 16(8).
Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal Meteorological Society, 81(348), 158-171.
Kanagawa, A., Arizono, I., & Ohta, H. (1997). Design of the control chart based on Kullback--Leibler information. Frontiers in Statistical Quality Control, 183--192.
Kończak, G. (2013). On the use of the extreme value distribution in monitoring production processes. Acta Universitatis Lodziensis. Folia Oeconomica, 286, 307-315.
Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79-86.
Kullback, S. (1978). Information theory and statistics. New York: Dover Publications.
Kupperman, M. (1956). Further applications of information-theory to multivariate-analysis and statistical-inference. Annals of Mathematical Statistics, 27(4), 1184.
Landwehr, J. M., Matalas, N. C., & Wallis, J. R. (1979). Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Research, 15(5), 1055-1064.
Mahdi, S., & Cenac, M. (2005). Estimating parameters of Gumbel distribution using the methods of moments, probability weighted moments and maximum likelihood. Revista de Matemática: Teoría y Aplicaciones, 12(1-2), 151.
Nawa, V., & Nadarajah, S. (2024). Exact expressions for Kullback–Leibler divergence for univariate distributions. extit{Entropy}, 26(11), 959.
Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1), 100-115.
Perry, M. B. (2020). An EWMA control chart for categorical processes with applications to social network monitoring. Journal of Quality Technology, 52(2), 182-197.
Pongpullponsak, A., & Intaramo, R. (2012). Development of fuzzy extreme value theory control charts using 𝛼-cuts for skewed populations. Applied Mathematical Sciences, 6(117), 5811–5834.
Reynolds Jr, M. R., & Stoumbos, Z. G. (1999). A CUSUM chart for monitoring a proportion when inspecting continuously. Journal of Quality Technology, 31(1), 87-108.
Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239-250.
Rosaiah, K., Rao, B. S., Reddy, J. P., & Chinnamamba, C. (2018). Variable control charts for Gumbel distribution based on percentiles. Journal of Computer and Mathematical Sciences, 9(12), 1890-1897.
Sarmadi, H., & Yuen, K.-V. (2022). Structural health monitoring by a novel probabilistic machine learning method based on extreme value theory and mixture quantile modeling. Mechanical Systems and Signal Processing, 173.
Shah, M. T., Azam, M., Hanif, M., Aslam, M., & Sherazi, U. (2023). Monitoring largest extreme observations using Fréchet distribution based on weighted variance method. Communications in Statistics—Theory and Methods, 52(22), 8136-8151.
Spliid, H. (2010). An exponentially weighted moving average control chart for Bernoulli data. Quality and Reliability Engineering International, 26(1), 97-113.
Sun, J., & 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.
Toshkova, D., Asher, M., Hutchinson, P., & Lieven, N. (2020). Automatic alarm setup using extreme value theory. Mechanical Systems and Signal Processing, 139.
Vermaat, T. M. B., Ion, R. A., Does, R. J. M. M., & Klaassen, C. A. J. (2003). A comparison of Shewhart individuals control charts based on normal, non‐parametric, and extreme‐value theory. Quality and Reliability Engineering International, 19(4), 337-353.
Villeta, M., Valencia, J. L., Saá, A., & Tarquis, A. M. (2017). Evaluation of extreme temperature events in northern Spain based on process control charts. Theoretical and Applied Climatology, 131(3–4), 1323-1335.
Yeh, A. B., Mcgrath, R. N., Sembower, M. A., & 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.