識別序列數據中的變化，稱為改變點偵測，已成為各個領域越來越重要的話
題。改變點偵測法可以分為即時和線下，我們主要針對即時改變點的方法做研究與推廣，稱之為 EXact Online Bayesian Changepoint Detection (EXO)，已經對於真實資料顯示出合理的結果。其中，對於資料型態，EXO 假設資料點間是相互獨立的，在真實資料中，資料間其實是有一定的相關性的，對於這種有相關性的資料，我們使用 Clayton copula 之下的馬可夫鏈模型，邊際分配的部分我們使用卜瓦松去描述這種間斷型的資料。從模擬得知在強相關性的情況下，這個模型有較好的準確性。並在實證資料中這個模型與 EXO 方法得到相同的結果。

Detecting the structure change in sequential data, known as changepoint detection,has become increasingly important in various fields. As the changepoint detection method
can be categorized by online and offline, this research focuses on the online way called EXact Online Bayesian Changepoint Detection (EXO). However, EXO assumes that the
datapoints are independent of each other, but this may be unrealistic. For real data, there is a certain relation between the datapoints. Therefore we consider the Markov chain model under the Clayton copula with the Poisson distribution as the marginal distribution to describe the data with the dependence structure and illustrate the performance in simulation studies. The data analysis comes from empirical studies.

Adams, R.P. and MacKay, D.J. (2007). Bayesian online changepoint detection. https://
doi.org/10.48550/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.
Byrd, M., Nghiem, L. and Cao, J. (2017). Lagged exact Bayesian online changepoint detection with parameter estimation. https://doi.org/10.48550/arXiv.1710.03276.
Carlin, B.P., Gelfand, A.E. and Smith, A.F. (1992). Hierarchical Bayesian analysis of changepoint problems. Journal of the Royal Statistical Society: Series C (Applied Statistics), 41,
389-405.
Chib, S. (1998). Estimation and comparison of multiple change-point models. Journal of
econometrics, 86, 221-241.
Chen, X. and Fan, Y. (2006). Estimation and model selection of semiparametric copulabased multivariate dynamic models under copula misspecification. Journal of Econometrics, 135, 125-154.
Clayton, D.G. (1978). A model for association in bivariate life tables and its application in
epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65,
141–1.
Chen, C.W.S. and Lee, B. (2021). Bayesian inference of multiple structural change models
with asymmetric GARCH errors. Statistical Methods and Applications, 30, 1053-1078.
Darsow, W.F., Nguyen, B. and Olsen, E.T. (1992). Copulas and Markov processes. Illinois
Journal of Mathematics, 36, 600-642.
Domma, F., Giordano, S. and Perri, P.F. (2009). Statistical modeling of temporal dependence
in financial data via a copula function. Communications in Statistics-Simulation and Computation, 38, 703-728.
Emura, T. and Chen, Y.H. (2018). Analysis of survival data with dependent censoring:
Copula-Based Approaches. Springer, Singapore.
Emura, T., Lai, C.C. and Sun, L.H. (2021). Change point estimation under a copula-based
Markov chain model for binomial time series. Econometrics and Statistics.
Fearnhead, P. and Liu, Z. (2007). On‐line inference for multiple changepoint problems. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69, 589-605.
Genest, C. and MacKay, R.J. (1986). Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données. The Canadian Journal of Statistics, 14, 145-159.
Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian
model determination. Biometrika, 82, 711-732.
Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
Joe, H. (1997). Multivariate models and multivariate dependence concepts. CRC press,
Florida.
Long, T.H. and Emura, T. (2014). A control chart using copula-based Markov chain models.
Journal of the Chinese Statistical Association, 52, 466–496.
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 (2nd ed.). Springer Series in Statistics.
Berlin.
Perry, M.B. and Pignatiello, J.J.Jr. (2005). Estimation of the change point of the process
fraction nonconforming in SPC applications. International Journal of Reliability, Quality
and Safety Engineering, 12, 95-110.
Reeves, J., Chen, J., Wang, X.L., Lund, R. and Lu, Q.Q. (2007). A review and comparison
of changepoint detection techniques for climate data. Journal of Applied Meteorology and
Climatology, 46, 900–915.
Shields, A., Doody, P. and Scully, T. (2017). Application of multiple change point detection
methods to large urban telecommunication networks. 2017 28th Irish Signals and Systems
Conference (ISSC), 1-6.
Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publications de l＇
Institut de Statistique de l＇Université de Paris, 8, 229–231.
Sun, L.H., Huang, X.W., Alqawba, M.S., Kim, J.M. and Emura, T. (2020). Copula-based
Markov models for time series: Parametric inference and process control. Springer Nature,
Berlin.