尺度矩阵的级数展开式及其在变点检测中的应用
A series expansion formula of the scale matrix with applications in change-point detection
摘要：
我们引入了一种新的Levy涨落理论方法来分析序列变点检测中的累积和（Cumulative sum，CSUM）过程。当观测值服从位相型分布，且变化后分布由其变化前分布的指数倾斜给出时，可将观测值的首达分析归结为一个马尔可夫可加过程的首达分析.我们开发了一个新的级数展开公式的规模矩阵的马尔可夫加性过程的有限活动，并应用它来获得精确的表达式的平均游程长度，平均检测延迟，和虚警概率下的CRACUUM程序。
英文摘要：
We introduce a new Levy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.
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