Structural Preservation in Time Series Through Multiscale Topological Features Derived from Persistent Homology

A principled, model-agnostic framework for structural feature extraction in time series is presented, grounded in topological data analysis (TDA). The motivation stems from two gaps identified in the literature: First, compact and interpretable representations that summarise the global geometric organisation of trajectories across scales remain scarce. Second, a unified, task-agnostic protocol for evaluating structure preservation against established non-topological families is still missing. To address these gaps, time-delay embeddings are employed to reconstruct phase space, sliding windows are used to generate local point clouds, and Vietoris–Rips persistent homology (up to dimension two) is computed. The resulting persistence diagrams are summarised with three transparent descriptors—persistence entropy, maximum persistence amplitude, and feature counts—and concatenated across delays and window sizes to yield a multiscale representation designed to complement temporal and spectral features while remaining computationally tractable. A unified experimental design is specified in which heterogeneous, regularly sampled financial series are preprocessed on native calendars and contrasted with competitive baselines spanning lagged, calendar-driven, difference/change, STL-based, delay-embedding PCA, price-based statistical, signature (FRUITS), and network-derived (NetF) features. Structure preservation is assessed through complementary criteria that probe spectral similarity, variance-scaled reconstruction fidelity, and the conservation of distributional shape (location, scale, asymmetry, tails). The study is positioned as an evaluation of representations, rather than a forecasting benchmark, emphasising interpretability, comparability, and methodological transparency while outlining avenues for adaptive hyperparameter selection and alternative filtrations.