Spectral SNR-t Bias Correction for Diffusion Models in Non-Stationary Time Series Forecasting

Diffusion probabilistic models have been increasingly applied to probabilistic time series forecasting. However, the inference-time signal-to-noise ratio–timestep (SNR-$t$) bias previously observed in visual diffusion models remains insufficiently explored in this setting. SNR-$t$ bias denotes an inference-time mismatch between the empirical SNR of an intermediate denoising sample and the expected SNR implied by its nominal diffusion timestep. While training establishes a fixed correspondence between timesteps and noise statistics through the forward perturbation process, accumulated reverse-process errors can break this correspondence during inference. In this work, we systematically examine SNR-$t$ bias in diffusion-based time series forecasting across multiple models and benchmark datasets. We find that the bias exhibits a distinctive spectral signature: the estimated noise residual develops frequency-dependent power distortions along the reverse trajectory, a phenomenon we term **spectral SNR-$t$ bias**. This spectral distortion can introduce spurious trends, periodic artifacts, and miscalibrated prediction intervals. Building on this diagnosis, we propose **Spectral-Shape SNR Correction (SSSC)**, a training-free, plug-and-play inference procedure that adaptively reshapes intermediate samples toward the spectral profile expected at their nominal timesteps. SSSC uses spectral energy to modulate frequency-wise correction strengths calibrated on held-out data and abstains from correction when reliable calibration is not possible. Experiments across seven diffusion forecasters show that SSSC is particularly effective for over-dispersed forecasts, reducing CRPS by up to $55\%$ relative to uncorrected baselines while also improving point forecast accuracy and probabilistic calibration.