Multi-stream sequential change detection involves simultaneously monitoring many streams of data and trying to detect when their distributions change, if at all. Here, we theoretically study multiple testing issues that arise from detecting changes in many streams. We point out that any algorithm with finite average run length (ARL) must have a trivial worst-case false detection rate (FDR), family-wise error rate (FWER), and per-family error rate (PFER); thus, any attempt to control these Type I error metrics is fundamentally in conflict with the desire for a finite ARL. One of our contributions is to define a new class of metrics which can be controlled, called error over patience (EOP). We propose algorithms that combine the recent e-detector framework (which generalizes the Shiryaev-Roberts and CUSUM methods) with the recent e-Benjamini-Hochberg procedure and e-Bonferroni procedures. We prove that these algorithms control the EOP at any desired level under very general dependence structures on the data within and across the streams. In fact, we prove a more general error control that holds uniformly over all stopping times and provides a smooth trade-off between the conflicting metrics. Additionally, if finiteness of the ARL is forfeited, we show that our algorithms control the Type I error.

翻译：多流序贯变点检测涉及同时监测多个数据流，并试图检测其分布是否发生改变以及何时发生改变。本文从理论上研究了在检测多个数据流变化时出现的多重检验问题。我们指出，任何具有有限平均运行长度（ARL）的算法，其最坏情况下的错误发现率（FDR）、族错误率（FWER）和每族错误率（PFER）必然是平凡的；因此，任何控制这些第一类错误度量的尝试本质上都与获得有限ARL的期望相冲突。我们的贡献之一是定义了一类新的、可以控制的度量，称为耐心误差（EOP）。我们提出了将最近的e-detector框架（该框架推广了Shiryaev-Roberts和CUSUM方法）与最近的e-Benjamini-Hochberg程序以及e-Bonferroni程序相结合的算法。我们证明，在数据流内部及数据流之间非常一般的依赖结构下，这些算法能在任何期望水平上控制EOP。事实上，我们证明了一个更一般的误差控制，该控制对所有停时一致成立，并在相互冲突的度量之间提供了平滑的权衡。此外，如果放弃ARL的有限性，我们证明了我们的算法能够控制第一类错误。