Paloma: Phase-Conditioned Residual Modulation for Time Series Forecasting

Explicitly modeling periodicity has proven to be an efficient approach for time series forecasting, where the residual, obtained after removing the periodic component from the time series, is typically treated as unstructured noise. However, in this work, we show that such residuals still exhibit structured, \textit{phase-dependent} deviations, as supported by both empirical observations and theoretical analysis. This suggests that residuals should not be treated merely as unstructured noise, but instead as \textit{phase-conditioned signals} whose variations arise in both direction and magnitude. Motivated by this perspective and inspired by phase and amplitude modulation in classical signal processing, we propose \textit{Paloma}, a simple yet effective framework for modeling residual deviations. Specifically, Paloma comprises two complementary modules: \textit{a Phase Rotation module}, which performs phase-conditioned rotations to capture directional changes in the feature space, and \textit{an Amplitude Scaling module}, which applies phase-conditioned affine transformations to model magnitude variations. We further abstract the two modules into a general-purpose plugin, termed the \textit{Paloma technique}, which can be readily integrated into existing forecasting architectures to modulate input sequences in a phase-aware manner, enabling backbone models to capture phase-dependent variations in both direction and magnitude. Extensive experiments on twelve datasets demonstrate that Paloma achieves state-of-the-art and robust forecasting performance. Additional results show that the Paloma technique consistently improves a wide range of baseline models, validating its effectiveness as a general-purpose component.