當生成資料序列的參數發生異常的改變，我們稱之為改變點，而改變點偵測就是為了偵測參數不規則變化的發生。改變點偵測又分即時偵測與線下偵測，這篇論文主要聚焦在即時改變點偵測，而即時改變點偵測是當我們隨著時間得到觀察值的同時判斷該觀察值是否為改變點。這個方法最主要的目標就是在改變點出現之後，能盡快發現改變點，並做出相對應的策略，在理想的狀況下，我們希望能在改變點出現的同時就能偵測到改變點的訊號。在此工作中，我們利用分析步長來決定可能的改變點。此外，在實務中，由於獨立的假設在大部分序列型資料是不成立的，我們考慮馬可夫鏈模型去描述資料之間的相關性。並且使用有左尾相關性的Clayton copula與右尾相關性
的Joe copula來描述聯合分配，邊際分配的部分為二項分配。在實證分析中，珠寶製造的數據為分析資料，目的是盡早找出改變點的發生，並檢查製造珠寶的過程，以降低製造珠寶的不良率。

When the parameters of the marginal distribution for the sequential data are abnormally changed, we call it the changepoint, and changepoint detection is to detect
the occurrence of irregular changes in parameters. Changepoint detection is divided into online detection and offline detection. This paper mainly focuses on online changepoint detection. Online changepoint detection is to monitor whether the observation is a changepoint or not when we obtain data over time. The purpose is to detect the changepoint as soon as possible after the changepoint appears, and make corresponding strategies. We
focus on fraction nonconforming change by using the most possible run length to identify changepoint in this paper. We compute the probability of the run length distribution to
identify the most possible run length at each time. In addition, since it is not practical to assume the data is independent in most cases, we describe the dependence structure by using the copula-based Markov model. In this work, we consider the Clayton copula and
the Joe copula and the marginal distribution is the binomial distribution. In empirical analysis, jewelry manufacturing data analysis is for the purpose of finding out when the
changepoint occurs as early as possible and checking the process of manufacturing the jewelry, so as to reduce the probability of broken jewelry.

References

Abberger, K. (2005). A simple graphical method to explore tail-dependence in stock-return
pairs, Applied Financial Economics, 15, 43-51.
Adams, R.P. and Mackay, D.J. (2007). Bayesian online changepoint detection. arXiv
preprint arXiv: 0710.3742

Aminikhanghahi, S. and Cook, D.J. (2017). A survey of methods for time series change
point detection. Knowledge and Information Systems, 51, 339–367.
Assareh, H., Smith, I. and Mengersen, K. (2015). changepoint detection in risk adjusted
control charts. Statistical Methods in Medical Reasearch, 24, 747-768.
Barry, D. and Hartigan J.A. (1992). Product partition models for changepoint problems.

The Annals of Statistics, 20, 260-279.
Burr, W.I. (1979). Elementary Statistical Quality Control. Dekker, New York.
Duran, R.I. and Albin, S.L. (2009). Monitoring a fraction with easy and reliable settings
of the false alarm rate. Quality and Reliability Engineering International, 25, 1026-1043
Emura, T. and Liao, Y.T. (2018). Critical review and comparison of continuity correction
methods: the normal approximation to the binomial distribution. Communication in
Statistics - Simulation and Computation, 47, 2266-2285.
Emura, T. and Lin, Y.S. (2015). A comparison of normal approximation rules for attribute
control charts. Quality and Reliability Engineering International, 31, 411-418.
Emura, T., Lai, C.C., and Sun, L.H. (2021). Changepoint estimation under a copula-based
Markov chain model for binomial time series. Econometrics and Statistics.
Emura, T., Long, T.H. and Sun, L.H. (2017). R routines for performing estimation and
statistical process control under copula-based time series models, Communications in
Statistics-Simulation and Computation, 46, 3067-3087.
Fisher, N.I. and Switzer, P. (1985). Chi-plots for assessing dependence, Biometrika, 72,
253-265.
Genest, C. and MacKay, R.J. (1986). Copules archim´ediennes et families de lois bidimensionnelles
dont les marges sont donn´ees, Canadian Journal of Statistics, 14, 145-159.
Johnson, T.D., Elashoff, R.M. and Harkema, S.J. (2003). A Bayesian changepoint analysis
of electromyographic data: detecting muscle activation patterns and associated
applications. Biostatistics, 4, 143-164.
Kendall, M.G. (1938). A new measure of rank correlation. Biometrika, 30, 81-93.
Lund, R., Reeves, J. (2002). etection of undocumented changepoints: A revision of the
two-phase regression model.Journal of Climate, 15, 2547-2554.
Montgomery, D.C. (2009). Introduction to Statistical Quality Control. Wiley, New York.
Nagaraj, N.K. (1990). Two-sided tests for change in level for correlated data. Statistical
Papers, 31, 181-194.
Nelsen, R.B. (2006). An Introduction to Copulas. Springer, New York.
Perry, M.B., Pignatiello, J.J. and Simpson, J.R. (2007). Estimating the changepoint of the
process fraction nonconforming with a monotonic change disturbance in SPC. Equality
and Reliability Engineering International, 23, 327-339.
Pignatiello, J.J. and Samuel, T.R. (2000). Identifying the time of a step change in the
process fraction nonconforming. Quality Engineering, 13, 357-365.
Ritov, Y., Raz, A., Bergman, H. (2002). Detection of onset of neuronal activity by
allowing for heterogeneity in the change points. Journal of neuroscience methods, 122,
25-42.
Rossi, G., Sarto, S.D., and Marchi, M.A. (2014). New risk-adjusted Bernoulli cumulative
sum chart for monitoring binary health data. Statistical Methods in Medical Reasearch.
doi: 10.1177/0962280214530883.

Sklar, A. (1959). Fonctions de re’partition a’n dimensions et leurs marges, Publica-tions
de l’Intitut de Statistique de l’Universit de Paris, 8, 229-231.
Wang, H. (2009). Comparison of p control charts for low defective rate. Computational
Statistics and Data Analysis, 53, 4210-4220.
Wetherill, G.B. and Brown, D.W. (1911). Statistical Process Control: Theory and Practice.
Chapman and Hall, London.