We propose an efficient online kernel Cumulative Sum (CUSUM) method for change-point detection that utilizes the maximum over a set of kernel statistics to account for the unknown change-point location. Our approach exhibits increased sensitivity to small changes compared to existing methods, such as the Scan-B statistic, which corresponds to a non-parametric Shewhart chart-type procedure. We provide accurate analytic approximations for two key performance metrics: the Average Run Length (ARL) and Expected Detection Delay (EDD), which enable us to establish an optimal window length on the order of the logarithm of ARL to ensure minimal power loss relative to an oracle procedure with infinite memory. Such a finding parallels the classic result for window-limited Generalized Likelihood Ratio (GLR) procedure in parametric change-point detection literature. Moreover, we introduce a recursive calculation procedure for detection statistics to ensure constant computational and memory complexity, which is essential for online procedures. Through extensive experiments on both simulated and real data, we demonstrate the competitive performance of our method and validate our theoretical results.

翻译：我们提出了一种高效的在线核累积和（CUSUM）方法用于变点检测，该方法利用一组核统计量的最大值来处理未知变点位置的问题。与现有方法（如对应非参数休哈特控制图型过程的Scan-B统计量）相比，我们的方法对小变化表现出更高的灵敏度。我们为两个关键性能指标——平均运行长度（ARL）和期望检测延迟（EDD）——提供了精确的解析近似，这使得我们能够建立最优窗口长度（与ARL的对数同阶），以确保相对于具有无限记忆的基准程序，功率损失最小。这一发现与参数变点检测文献中关于有限窗口广义似然比（GLR）过程的经典结果相呼应。此外，我们引入了一种检测统计量的递归计算过程，以保证常数时间和空间复杂度，这对在线过程至关重要。通过在模拟数据和真实数据上的大量实验，我们证明了该方法具有竞争性性能，并验证了我们的理论结果。