When data goes missing: a topological perspective of sampling rate and missing data in nonlinear time series analysis

Abstract Reconstructing attractors from nonlinear time series is central to nonlinear dynamics, particularly when only a single observable is available. Although Takens’ embedding theorem guarantees diffeomorphic reconstruction in theory, practical issues such as the role of sampling frequency and missing data still remain unexplored. In this work, we study how these factors affect reconstruction quality from a topological perspective. Hence, in our work, we consider two scenarios: a) We vary the sampling frequency of the time series data b) Time series data with varied percentages of missing points (scanty data). For each case, we reconstruct the attractor using optimal embedding parameters and evaluate the topology via persistent homology (PH). Their fidelity is quantified through various measures, such as the behavior of optimal parameters, the persistence of true features with respect to topological noise, the area under the curve (AUC) analysis of Betti curves extracted from PH, and comparing all these measures using the original phase space as a reference. We tested our methodology on two widely studied systems: Duffing and Lorenz. Our results show that sequential sampling (with different sampling frequencies) has a negligible impact on the disruption of the topological invariants. In contrast, scanty data (random removals) leads to strong variability in AUC of Betti curves, severe degradation of reconstruction quality, and disruption of topological invariants (true features), making it lose the dynamical information. This study highlights a crucial insight: even minimal data loss can severely hamper the reconstruction fidelity. These findings underscore the importance of reliable and continuous sensor recordings in experimental settings, as seen in data-driven studies in nonlinear physics, physiology, and climate science.