The problem of sequential anomaly detection and identification is considered in the presence of a sampling constraint. Specifically, multiple data streams are generated by distinct sources and the goal is to quickly identify those that exhibit ``anomalous'' behavior, when it is not possible to sample every source at each time instant. Thus, in addition to a stopping rule, which determines when to stop sampling, and a decision rule, which indicates which sources to identify as anomalous upon stopping, one needs to specify a sampling rule that determines which sources to sample at each time instant. The focus of this work is on ordering sampling rules, which sample the data sources, among those currently estimated as anomalous (resp. non-anomalous), for which the corresponding local test statistics have the smallest (resp. largest) values. It is shown that with an appropriate design, which is specified explicitly, an ordering sampling rule leads to the optimal expected time for stopping, among all policies that satisfy the same sampling and error constraints, to a first-order asymptotic approximation as the false positive and false negative error rates under control both go to zero. This is the first asymptotic optimality result for ordering sampling rules when multiple sources can be sampled per time instant. Moreover, this is established under a general setup where the number of anomalies is not required to be a priori known. A novel proof technique is introduced, which unifies different versions of the problem regarding the homogeneity of the sources and prior information on the number of anomalies.

翻译：在采样约束下，考虑序列异常检测与识别问题。具体而言，多个数据流由不同源生成，目标是在无法每个时刻对所有源进行采样时，快速识别表现出“异常”行为的源。因此，除确定何时停止采样的停止规则和指示停止后哪些源被识别为异常的决策规则外，还需指定确定每个时刻对哪些源进行采样的采样规则。本研究聚焦于序采样规则——该规则在当前被估计为异常（或非异常）的数据源中，优先采样对应局部检验统计量值最小（或最大）的源。研究表明，通过明确指定的合适设计，在控制假阳性和假阴性错误率趋于零的一阶渐近逼近下，序采样规则能在满足相同采样和错误约束的所有策略中达到最优期望停止时间。这是首个针对单时刻可采样多个源时序采样规则的渐近最优性结果。此外，该结论在无需预先已知异常数量的通用框架下成立。本文引入了一种新颖的证明技术，统一了关于源同质性和异常数量先验信息的不同版本问题。