根據觀察到的序列數據，如果存在一個點將數據從一個特定分佈轉變為另一
個分佈，則該點稱為改變點。改變點檢測可以幫助我們事前警惕和事後分析。
有幾種方法可以檢測改變點。在傳統方法中，最大概似估計法可以幫助我們估
計獨立觀測序列的改變點。然而，在實際數據中，觀測值通常不是獨立的，這
可能導致使用傳統的最大概似估計法獲得準確的改變點估計存在困難。因此，
我們研究了 Clayton copula 下的馬爾可夫鏈，其中邊際分佈是混合正態分佈，以獲得非獨立觀測值下的改變點估計值。此外，模型構建了一個新的最大概似函數，並通過牛頓-拉弗森方法求解參數。通過實證研究來展示所提出方法的性能。

Based on the observed sequential data, if there is structural change at some particular time, this critical point of structural change is the change point. The change point detection helps in both proactive warning and retrospective analysis. In real applications, observations are often not independent. Therefore, we propose the Clayton copula based the Markov chain which the marginal distribution is the mixture normal distribution to estimate the change point for the time dependent observations. Consequently, the corresponding maximum likelihood function is obtained and the maximum likelihood estimates are solved through the Newton-Raphon method. The performance for the proposed method is illustrated using simulation studies. Finally, we discuss the structural change in the real application based on two periods: 2008 financial crisis and 2020 COVID-19.

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