TopoCast: A Topological Fidelity Framework for Evaluating Transformer-Based Time Series Forecasting

Abstract

Deep learning-based models have achieved state-of-the-art performance in Time Series Forecasting (TSF), yet their evaluation remains dominated by pointwise error metrics such as Mean Squared Error (MSE), which quantify numerical accuracy but overlook structural properties of the forecast signal, including recurrent dynamics, oscillatory behavior, and phase alignment. As a result, forecasts exhibiting over-smoothing, phase shifts, or frequency distortions may achieve favorable error scores despite substantial structural degradation. To address this limitation, we propose TopoCast, a topology-driven framework for evaluating structural fidelity in TSF. TopoCast reconstructs phase-space representations of forecast and ground-truth sequences using Takens delay embedding and applies persistent homology to characterize their intrinsic dynamics. We derive four complementary topological fidelity measures from persistence diagrams and aggregate them into a Topological Fidelity Score (TFS). We further introduce dominant cycle overlap, a novel metric that maps persistent topological features to the temporal domain to assess whether dominant oscillatory patterns occur at the correct time points. Combined with TFS, this yields the Localized Topological Fidelity Score (LTFS), a phase-aware measure that captures temporal localization errors invisible to existing evaluation metrics. Experiments on five Transformer architectures across three real-world benchmark datasets demonstrate that models with similar forecasting errors can exhibit markedly different structural fidelity profiles, revealing failure modes overlooked by conventional evaluation and highlighting the value of topology-aware forecast assessment.