🤖 AI Summary
This work proposes a novel method for online change-point detection in high-dimensional nonlinear time series by integrating persistent homology with Laplacian spectra to simultaneously control false alarms and reduce detection delay. The approach maps sliding windows of the time series into point clouds and leverages persistent Laplacian spectra—going beyond conventional homology-based counts—to capture scale-dependent geometric structures and connectivity. Embedded within a recursive Page-CUSUM monitoring framework, the method enables efficient real-time surveillance. By incorporating whitened scores and a two-phase (Phase I/II) parameter calibration procedure, it provides theoretical guarantees for false alarm control over finite monitoring horizons. Empirical evaluations on both synthetic and real-world datasets demonstrate consistently reliable false alarm rates and state-of-the-art detection performance.
📝 Abstract
We propose the Persistent Laplacian Cumulative Sum (PL-CUSUM), an online change-point detection method for high-dimensional nonlinear time series. The method converts sliding windows into point clouds and uses persistent Laplacian spectra to construct the monitoring score for the Page cumulative sum (Page-CUSUM) recursion. Compared with detectors based only on persistent-homology summaries, PL-CUSUM further uses spectral information to capture within-scale connectivity and geometric structure beyond homology counts. Theoretically, we analyze two key performance criteria: false-alarm control and detection delay. We derive false-alarm-delay bounds for the oracle detector and show that the plug-in whitened score still controls false alarms over a finite monitoring horizon. Methodologically, we provide a Phase I/Phase II procedure that performs parameter selection and control-limit calibration before online recursion. Experiments on simulated systems and real monitoring data show that PL-CUSUM provides stable false-alarm control and competitive detection performance.