Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals
Recurrent signals give rise to trajectories that repeatedly return close to earlier states in state space. Many analysis methods therefore require a principled notion of similarity between states. In practice, a recurrence threshold sets the scale of the neighbourhood used to define when two states are considered close. Close returns can also support topology-preserving denoising in state space, aiming to reduce noise while preserving the trajectory's structure, which classical denoising methods may distort. The effectiveness of both denoising and recurrence analysis therefore depends critically on how these neighbourhoods are modelled and scaled. This work introduces a flow-aware ellipsoidal filtration for persistent homology based on a spatio--temporal covariance construction that estimates local flow geometry from both temporal and spatial neighbours. Unlike isotropic constructions based on balls (e.g.\ the Vietoris--Rips filtration), the proposed method assigns an ellipsoid to each point, with orientation and axis lengths determined by local flow variances. When a dominant $H_1$ feature reflects the recurrent loop structure, its persistence interval provides a data-driven scale selection. Across the considered experiments, flow-aware ellipsoidal neighbourhoods improve topology-preserving denoising and first-recurrence-time estimation relative to the Vietoris--Rips filtration. Overall, the results indicate that persistent homology can be more informative for dynamical systems when domain knowledge is used to incorporate anisotropy.