We consider online monitoring of the network event data to detect local changes in a cluster when the affected data stream distribution shifts from one point process to another with different parameters. Specifically, we are interested in detecting a change point that causes a shift of the underlying data distribution that follows a multivariate Hawkes process with exponential decay temporal kernel, whereby the Hawkes process is considered to account for spatio-temporal correlation between observations. The proposed detection procedure is based on scan score statistics. We derive the asymptotic distribution of the statistic, which enables the self-normalizing property and facilitates the approximation of the instantaneous false alarm probability and the average run length. Thus, when detecting a change in the Hawkes process with non-vanishing self-excitation, the procedure does not require estimating the post-change network parameter while assuming the temporal decay parameter. We further present an efficient procedure to accurately determine the false discovery rate via importance sampling, as validated by numerical examples. The good performance of our procedures compared with the benchmarks is tested with numerical experiments with simulated and real stock exchange data.
翻译:我们考虑对网络事件数据进行在线监测,以便在受影响的数据流分布过程从一个点转向另一个点,并带有不同的参数时,发现一个组群的局部变化。具体地说,我们有兴趣发现一个变化点,导致基本数据分布随着指数衰减时间内核的多变的霍克斯过程而发生变化,根据这个变化点,霍克斯过程被认为考虑到观测之间的时空关系。提议的检测程序以扫描分统计为基础。我们从统计的无规律分布中得出统计数据,使属性能够自我正常化,便于近似瞬时假警报概率和平均运行长度。因此,当用非加速自我刺激的方式发现霍克斯过程的变化时,程序不需要在假设时间衰减参数的同时估计变化后的网络参数。我们进一步提出一种高效的程序,以便通过重要抽样来准确确定虚假发现率,并用数字实例加以验证。我们程序与基准相比的良好表现通过模拟和实际的库存交换数据进行数字试验。