Change Point Detection, Localization, and Imputation of Topological Signals

Abstract

We propose a topology-aware online framework for detecting and localizing change points in partially observed temporal signals defined over cellular complexes. We model the process as a linear state space model. The latent dynamics follow a topology-consistent stochastic partial differential equation (SPDE), and the observations a topologically-filtered version of the state. Hidden states are inferred via a Kalman filter, while model parameters are adapted online through likelihood-based updates. Change points are detected by tracking deviations in the uncertainty parameters using a score-based criterion, and the responsible components are localized by mapping these deviations to specific cells in the complex. Under missing or noisy observations, backward Kalman smoothing—reinitialized at each detected change—provides consistent state reconstruction and reliable imputation. Experiments on synthetic and EPANET-simulated water networks demonstrate accurate detection, precise localization, and robust reconstruction under partial observability.