布阿松迴歸(Poisson Regression)是一種使用於計數型資料(Count Data)的迴歸分析模型，此模型已被廣泛應用於保險、生物、經濟、醫學及工程等領域。
在群集分析(Clustering Analysis)的領域，當資料具備某種機率分配特性時，EM 演算法是最常被使用，也是公認最實用且有效估計最大概似估計(Maximum Likelihood Estimate)的方法。
一般來說，改變點迴歸模型(Change-Point Regression Models)是使用於探測資料中改變點的位置，類似於使用排序的資料分群，很適合運用群集的方法。然而，在改變點迴歸模型中，EM理論卻很少被應用於估計改變點的位置。因此，我們考慮應用EM理論估計有改變點的布阿松迴歸模型。
在本篇論文中，我們提出了一個新的方法叫做EM改變點迴歸演算法 (EM Change-Point Regression Algorithm)，應用於布阿松迴歸。經由假設改變點為未知的變數，透過極大化最大概似估計函數，我們應用EM改變點迴歸演算法(EMCPR)估計改變點的位置以及布阿松迴歸模型的參數。我們同時也提供了模擬資料測試結果以及真實資料分析。結果顯示EM改變點迴歸演算法(EMCPR)是一個有效且實用的改變點迴歸演算法。

關鍵字：EM 演算法, 改變點, 改變點迴歸模型, 布阿松迴歸模型, EM 改變點迴歸演算法

Poisson regression is a form of regression analysis used to model count data. These models have been widely applied in various fields, such as insurance, biology, economic, medical and engineering, etc.
The Expectation and Maximization (EM) algorithm is the most used method in clustering, when the data follows a known probability distribution. Furthermore, the EM algorithm is also acknowledged to the most practical and effective method to estimate the Maximum Likelihood Estimate (MLE).
Ordinarily, CP regression models deal with the problems of detecting the location of CPs which is similar to clustering with ordered data and is suitable for clustering methods. However, EM is less used in locating CPs in CP regression models. Therefore, we considered applying the EM algorithm to estimate change-points for Poisson regression models.
In this thesis, a new method called EM Change-Point regression (EMCPR) algorithm for Poisson mixture regressions is proposed. By assuming the Change-Points as unknown variables, we used the EMCPR algorithm to estimate the Change-Point and Poisson mixture regression parameters via maximizing the likelihood functions. We demonstrate the approach with several numerical examples and real datasets. The results show that the proposed EMCPR is an effective and useful CP regression algorithm.

Keywords: EM algorithm, Change-Point, Change-Point regression models, Poisson mixture regressions, EM Change-Point regression algorithm

References

[1] Paul J. Plummera and Jie Chen, “A Bayesian approach for locating change points in a compound Poisson process with application to detecting DNA copy number variations,” Journal of Applied Statistics, Vol. 41, No. 2, pp. 423–438, 2014
[2] Taeyoung Parka, Robert T. Krafty and Alvaro I. Sánchez, “Bayesian semi-parametric analysis of Poisson change-point regression models application to policy-making in Cali, Colombia,” Journal of Applied Statistics, Vol. 39, No. 10, pp. 2285–2298, 2012
[3] Marcus B. Perry, Joseph J. Pignatiello JR and James R. Simpson, “Change point estimation for monotonically changing Poisson rates in SPC,” International Journal of Production Research, Vol. 45, No. 8, pp. 1791–1813, 2007
[4] Elnaz Asghari Torkamani, Seyed Taghi Akhavan Niaki, Majid Aminnayeri, Mehdi Davoodi2, “Estimating the Change Point of Correlated Poisson Count Processes,” Quality Engineering, Vol. 26, pp. 182–195, 2014
[5] Seyed Taghi Akhavan Niaki and Majid Khedmati, “Monotonic change-point estimation of multivariate Poisson processes using a multi-attribute control chart and MLE,” International Journal of Production Research, Vol. 52, No. 10, pp. 2954–2982, 2014
[6] A. E. Raftery and V. E. Akman, “Bayesian Analysis of a Poisson Process with a Change-Point,” Biometrika, Vol. 73, No. 1, pp. 85–89, 1986
[7] Seyed Taghi Akhavan Niaki and Majid Khedmati, “Estimating the change point of the parameter vector of multivariate Poisson processes monitored by a multi-attribute T2 control chart,” Int J Adv Manuf Technol, Vol. 64, pp. 1625–1642, 2013
[8] Joseph G. Ibrahim, Ming-Hui Chen and Stuart R. Lipsitz, “Monte Carlo EM for Missing Covariates in Parametric Regression Models,” Biometrics, Vol. 55, No. 2, pp. 591–596, 1999
[9] Dimitris Karlis, “A general EM approach for maximum likelihood estimation in mixed Poisson regression models,” Statistical Modeling, Vol. 1, pp. 305–318, 2001
[10] Hossein Zamani, Pouya Faroughi and Noriszura Ismail, “Estimation of Count Data using Mixed Poisson, Generalized Poisson and Finite Poisson Mixture Regression Models,” AIP. Conf. Proc. 1602, pp. 1144–1150, 2014
[11] S. Faria and F. Gonçalves, “Financial data modeling by Poisson mixture regression,” Journal of Applied Statistics, Vol. 40, No. 10, pp. 2150–2162, 2013
[12] Peiming Wang, Martin L. Puterman, Iain Cockburn and Nhu Le, “Mixed Poisson Regression Models With Covariate Dependent Rates,” Biometrics, Vol. 52, No. 2, pp. 381–400, 1996
[13] Yanlin Tang, Liya Xiang, and Zhongyi Zhu, “Risk factor selection in rate making EM Adaptive LASSO for Zero-Inflated Poisson Regression Models,” Risk Analysis, Vol. 34, No. 6, pp. 1112–1127, 2014
[14] McLachlan, Geoffrey J., Krishnan, Thriyambakam, “The EM Algorithm and Extensions,” Chichester, Wiley, 1997.
[15] D. Karlis, “An EM algorithm for multivariate Poisson distribution and related models,” Journal of Applied Statistics, Vol. 30, No. 1, pp. 63–77, 2003
[16] Naveen K. Bansal, Hong Du and G. G. Hamendani, “An Application of EM Algorithm to a Change-Point Problem,” Communications in Statistics—Theory and Methods, Vol. 37 pp. 2010–2021, 2008
[17] Eichler, “Applications of MCMC,” May 8, 2003, http://galton.uchicago.edu/~eichler/stat24600/ Handouts/s10.pdf
[18] Edward N. Rappaport and Jose Fernandez-Partagas, “The Deadliest Atlantic Tropical Cyclones, 1492-1996,” 28 May 1995, http://www.nhc.noaa.gov/pastdeadly.shtml?
[19] Central Weather Bureau, “Earthquake Report, 2011-2016,” http://www.cwb.gov.tw/V7e/earthquake/quake_index.htm
[20] Society of Actuaries (SOA), “1991-92 Group Medical Insurance Large Claims Database,” November 2004, https://www.soa.org/Research/Experience-Study/group-health/91-92-group-medical-claims.aspx
[21] P. McCullagh and J.A. Nelder, Generalized Linear Models, 2nd Ed. CHAPMAN & HALL, 1989