CUSUM design for detection of event-rate increases for a Poisson process

In quantifying the performance of a CUSUM chart for detecting upward shifts in event rate, it has been recommended that steady-state evaluation of performance measures such as ARL be used. In this article, the methodology for making such evaluations using the Markov-chain approach is presented, for the case of an exponential CUSUM. This is much more efficient than the alternative of simulation, which is still in use. It is also shown that if one is using steady-state ARL as a measure of detection performance, one can find better choices for the CUSUM parameter <i>k</i> than that provided by the SPRT-based formula. Two types of shift in event rate are considered, and corresponding tables of recommended choices of CUSUM parameters (<i>k</i>, <i>h</i>) are presented for ten levels of in-control ARL, and for nine sizes of shift. These tables can assist quality engineers in the design of CUSUMs for monitoring inter-event times in steady-state operation. It is also shown that these exponential CUSUM tables may be used to find values for the parameters of a geometric CUSUM or a Bernoulli CUSUM chart for monitoring a proportion, provided the in-control value of the proportion is no more than approximately 0.5%.

在量化累积和（CUSUM）控制图用于检测事件率向上偏移的性能表现时，学界已建议采用稳态方式评估平均运行链长（ARL, Average Run Length）等性能指标。本文针对指数型累积和控制图场景，提出了基于马尔可夫链（Markov-chain）方法开展此类评估的完整方法论，该方案的效率远高于目前仍在广泛使用的仿真替代方法。研究同时表明，若以稳态平均运行链长作为检测性能的衡量标准，相较于基于序贯概率比检验（SPRT, Sequential Probability Ratio Test）的公式所给出的取值，可找到更优的累积和参数k选择方案。本文考虑了两类事件率偏移场景，并针对10种受控状态下平均运行链长水平、9种偏移幅度，给出了对应的累积和参数(k, h)推荐取值表。此类表格可辅助质量工程师设计用于稳态运行下事件间隔监测的累积和控制图。此外，研究证实，当受控比例的初始值不超过约0.5%时，可利用本文提出的指数型累积和控制图表，推导出用于监测比例的几何型累积和控制图或伯努利（Bernoulli）累积和控制图的参数取值。