STEPS: Spectral–Temporal Error Propagation for Test-Time Adaptation in Time-Series Forecasting

Test-time adaptation (TTA) for time-series forecasting must revise a forecast from a short prefix of delayed targets that may contain transient changes or corrupted measurements. Existing methods often gain more at long horizons than at short horizons and are sensitive to unreliable observations, as noisy or transient errors can be propagated into the unseen suffix. We observe that forecast errors exhibit two complementary structures: persistent cross-window bias is concentrated in low-frequency modes, while newly revealed residuals contain useful short-term structure whose influence on future errors must be selectively propagated rather than indiscriminately amplified. Based on these observations, we propose STEPS, a Spectral–Temporal Error Propagation Solver that adapts a frozen forecaster in output space. Multiscale Spectral Bias Reconstruction (MSBR) recovers persistent bias from historical residual spectra; Short-Term Temporal Error Propagation (ST-TEP) learns how the revealed prefix should influence the unseen suffix through an empirical temporal kernel; and Orthogonal Complementary Correction (OCC) removes their overlapping component before fusion. Across six datasets, seven backbones, and four horizons, STEPS delivers a 24.85% average MSE reduction, which is more than twice the gain achieved by the strongest TTA baseline. Under 1–30% Gaussian prefix corruption, STEPS significantly outperforms competing methods. These results demonstrate accurate and robust forecast revision across temporal scales without online backbone optimization.