PAMod: Modeling Cyclical Shifts via Phase-Amplitude Modulation for Non-stationary Time Series Forecasting

2026-05-01

AI Summary

Real-world time series often suffer degraded forecasting performance due to non-stationarity, such as time-varying means and variances, yet most existing methods rely on the strong assumption that historical and future distributions are identical. To address this limitation, this work proposes PAMod, a novel framework that models periodic distribution shifts as a phase-amplitude modulation mechanism in feature space: phase modulation captures dynamic mean changes, while amplitude modulation adapts to variance fluctuations. Theoretically, this mechanism is shown to be equivalent to dynamic de-normalization. PAMod is lightweight, plug-and-play, and integrates periodic embeddings with invertible instance normalization. Evaluated on twelve real-world benchmarks, it significantly outperforms state-of-the-art methods, consistently enhances the performance of mainstream models, and incurs lower computational overhead.

Abstract

Real-world time series forecasting faces the fundamental challenge of non-stationary statistical properties, including shifts in mean and variance over time. While reversible instance normalization (RevIN) has shown promise by stationarizing inputs and denormalizing outputs, it relies on the strong assumption that historical and future distributions remain identical. We observe that in many practical applications, distribution shifts follow cyclical patterns that correlate with periodic positions (e.g., seasonal and holiday volatility). To this end, we propose PAMod, a lightweight yet powerful framework that models cyclical distribution shifts via Phase-Amplitude Modulation in the normalized feature space. PAMod learns periodic embeddings to modulate representations: phase modulation captures mean shifts, while amplitude modulation adapts to variance changes. Crucially, we prove mathematically that modulating in normalized space is equivalent to applying dynamic denormalization, offering an elegant unification of distribution adaptation and representation learning. Extensive experiments on twelve real-world benchmarks demonstrate that PAMod achieves state-of-the-art performance with fewer computational resources. Furthermore, our modulation mechanism, as a novel plug-and-play technique, can improve existing time-series forecasting methods with simple integration.

Problem

• non-stationary time series
• distribution shift
• cyclical patterns
• time series forecasting
• statistical non-stationarity

Innovation

• Phase-Amplitude Modulation
• Non-stationary Time Series
• Cyclical Distribution Shifts
• Dynamic Denormalization
• Periodic Embeddings