本論文提出了貝氏在線變點檢測方法基於Clayton copula 且邊際分布為混合
常態分布的copula-Markov 模型。在模擬研究中，我們研究了相關時間序列數據
的結構變化，包括單峰數據和雙峰數據之間的轉換以及雙峰數據內部的結構變
化，並將其與邊際分布為常態分布的copula-Markov 模型進行比較，模擬的結果
指出在相關性資料由單峰改變成雙峰時，我們所使用的模型在準確率、平均絕對
誤差、真陽個數、偽陽個數皆勝過對比模型。我們應用這種方法來檢測加密貨幣
市場從2017 年到2018 年初的變點，其中包括ICO 繁榮的開始和結束，我們選用
BTC、ETH、NEO 的每日報酬率進行偵測。結果顯示了我們的方法在檢測相關時
間序列數據的變化方面的有效性，並提供了關於虛擬貨幣市場在這ICO 繁榮-蕭條
期間行為的見解。
關鍵字：在線改變點檢測，Clayton copula，馬可夫模型，貝氏推論，ICO boom

In this paper, we propose an online Bayesian changepoint detection approach for dependent
time series data under the copula-based Markov model with the marginal distributions
being mixture normal distributions. Simulation studies examine structural changes in
sequential time series data with dependency, including transformation between unimodal
and bimodal data and structural changes within bimodal data. For comparison, we consider
the Clayton copula-based Markov model with normal marginal distributions as the
benchmark model. The results show that the proposed model outperforms the benchmark
model in detecting change when the correlation structure of the changes from unimodal
data to bimodal data. In the empirical analysis, we use the daily returns of BTC, ETH, and
NEO to identify change points in the cryptocurrency market from 2017 to early 2018. The
results demonstrate the effectiveness of our approach in detecting changes and providing
insight into cryptocurrency market behavior during ICO booms and busts.
Keywords: online change point detection, Clayton copula, Markov model, Bayesian
inference, ICO boom

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.
Barry, D. and Hartigan, J. A. (1992). Product partition models for change point problems.
The Annals of Statistics, 20:260–279.
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,
65:141–151.
Darsow, W. F., Nguyen, B., and Olsen, E. T. (1992). Copulas and Markov processes.
Illinois Journal of Mathematics, 36:600 – 642.
Genest, C. and MacKay, J. (1986). The joy of copulas: Bivariate distributions with uniform
marginals. The American Statistician, 40:280–283.
Hastings, W. K. (1970). Monte carlo sampling methods using markov chains and their
applications. Biometrika, 57(1):97–109.
James, N., Menzies, M., and Chan, J. (2021). Changes to the extreme
and erratic behaviour of cryptocurrencies during covid-19.
Physica A: Statistical Mechanics and its Applications, 565:125581.
Kuo, D.-H. (2022). Online change point detection under a copula-based markov chain
model for normal sequence data.
Lavielle, M. and Lebarbier, E. (2001). An application of mcmc methods for the multiple
change-points problem. Signal Processing, 81:39–53. Special section on Markov Chain
Monte Carlo (MCMC) Methods for Signal Processing.
Lee, S. and Baek, C. (2023). Volatility changes in cryptocurrencies: evidence from sparse
vhar-mgarch model. Applied Economics Letters, 30(11):1496–1504.
Lin, W.-C., Emura, T., and Sun, L.-H. (2021). Estimation under copulabased
markov normal mixture models for serially correlated data.
Communications in Statistics - Simulation and Computation, 50:4483–4515.
McLachlan, G. J., Lee, S. X., and Rathnayake, S. I. (2019). Finite mixture models.
Annual Review of Statistics and Its Application, 6:355–378.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and
Teller, E. (1953). Equation of state calculations by fast computing machines.
Journal of Chemical Physics, 21:1087–1092.
Metropolis, N. and Ulam, S. (1949). The monte carlo method.
Journal of the American statistical association, 44:335–341.
Nelsen, R. B. (2007). An introduction to copulas. Springer science & business media.
Polunchenko, A. S. and Tartakovsky, A. G. (2012). State-of-the-art in sequential changepoint
detection. Methodology and computing in applied probability, 14(3):649–684.
Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges.
Publ. inst. statist. univ. Paris, 8:229–231.
Thies, S. and Molnár, P. (2018). Bayesian change point analysis of bitcoin returns.
Finance Research Letters, 27:223–227.
Truong, C., Oudre, L., and Vayatis, N. (2020). Selective review of offline change point
detection methods. Signal Processing, 167:107299.