SWiT: Spectral Wiener Correction for Test-Time Adaptation in Time Series Forecasting

Time-series forecasting models degrade after deployment because the statistical properties of the data shift over time in real-world environments. Test-time adaptation (TTA) keeps the deployed backbone frozen and updates a small adapter from the ground truth that arrives after each forecast. Because that ground truth arrives one step at a time, the freshest windows are only partially observed when adaptation runs, so an adapter must learn from incomplete labels without admitting any unobserved value. Existing methods answer this with modules updated by stochastic gradient descent, so incomplete labels can be used only through a gradient step; the correction depends on the learning rate and the number of steps taken, and the cost grows quadratically with the horizon. We propose Spectral Wiener correction for Test-time adaptation (simply, SWiT), which corrects a frozen backbone's forecast in the frequency domain and obtains that correction in closed form rather than by gradient descent. The prediction and its input history are transformed by the Fast Fourier Transform; every bin receives an affine correction (i.e., a scale, an input term, and a rotational bias phase-locked to absolute time), and the result is transformed back. By Parseval's theorem, the time-domain mean squared error (MSE) separates into independent per-bin quadratics, so the per-batch optimum is a closed-form Wiener-type solution over running sums, with no optimizer state or learning rate, at $\mathcal{O}(H\log H)$ cost in the forecasting horizon $H$. A causal protocol splices only observed steps, the prefix of each in-progress window, into those same running sums, so the freshest evidence enters the closed-form solve itself, and an evidence gate scales each bin's correction by the benefit it has delivered on past windows. Across 216 settings spanning six backbones, nine benchmarks, and four horizons, SWiT attains the lowest average MSE, $10.3\%$ below the frozen backbone and $5.3\sim 7.9\%$ below the state-of-the-art TTA methods, with $10\sim 257\times$ fewer parameters and the shortest adaptation time. The code is available at https://anonymous.4open.science/r/SWiT-808C.