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TimeAPN: Adaptive Amplitude-Phase Non-Stationarity Normalization for Time Series Forecasting

Yue Hu, Jialiang Tang, Siwei Yu, Baosheng Yu, Jing Zhang, Dacheng Tao

Abstract

Non-stationarity is a fundamental challenge in multivariate long-term time series forecasting, often manifested as rapid changes in amplitude and phase. These variations lead to severe distribution shifts and consequently degrade predictive performance. Existing normalization-based methods primarily rely on first- and second-order statistics, implicitly assuming that distributions evolve smoothly and overlooking fine-grained temporal dynamics. To address these limitations, we propose TimeAPN, an Adaptive Amplitude-Phase Non-Stationarity Normalization framework that explicitly models and predicts non-stationary factors from both the time and frequency domains. Specifically, TimeAPN first models the mean sequence jointly in the time and frequency domains, and then forecasts its evolution over future horizons. Meanwhile, phase information is extracted in the frequency domain, and the phase discrepancy between the predicted and ground-truth future sequences is explicitly modeled to capture temporal misalignment. Furthermore, TimeAPN incorporates amplitude information into an adaptive normalization mechanism, enabling the model to effectively account for abrupt fluctuations in signal energy. The predicted non-stationary factors are subsequently integrated with the backbone forecasting outputs through a collaborative de-normalization process to reconstruct the final non-stationary time series. The proposed framework is model-agnostic and can be seamlessly integrated with various forecasting backbones. Extensive experiments on seven real-world multivariate datasets demonstrate that TimeAPN consistently improves long-term forecasting accuracy across multiple prediction horizons and outperforms state-of-the-art reversible normalization methods.

TimeAPN: Adaptive Amplitude-Phase Non-Stationarity Normalization for Time Series Forecasting

Abstract

Non-stationarity is a fundamental challenge in multivariate long-term time series forecasting, often manifested as rapid changes in amplitude and phase. These variations lead to severe distribution shifts and consequently degrade predictive performance. Existing normalization-based methods primarily rely on first- and second-order statistics, implicitly assuming that distributions evolve smoothly and overlooking fine-grained temporal dynamics. To address these limitations, we propose TimeAPN, an Adaptive Amplitude-Phase Non-Stationarity Normalization framework that explicitly models and predicts non-stationary factors from both the time and frequency domains. Specifically, TimeAPN first models the mean sequence jointly in the time and frequency domains, and then forecasts its evolution over future horizons. Meanwhile, phase information is extracted in the frequency domain, and the phase discrepancy between the predicted and ground-truth future sequences is explicitly modeled to capture temporal misalignment. Furthermore, TimeAPN incorporates amplitude information into an adaptive normalization mechanism, enabling the model to effectively account for abrupt fluctuations in signal energy. The predicted non-stationary factors are subsequently integrated with the backbone forecasting outputs through a collaborative de-normalization process to reconstruct the final non-stationary time series. The proposed framework is model-agnostic and can be seamlessly integrated with various forecasting backbones. Extensive experiments on seven real-world multivariate datasets demonstrate that TimeAPN consistently improves long-term forecasting accuracy across multiple prediction horizons and outperforms state-of-the-art reversible normalization methods.
Paper Structure (24 sections, 22 equations, 3 figures, 4 tables)

This paper contains 24 sections, 22 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. RELATED WORK
  3. Time Series Forecasting
  4. Non-Stationarity Normalization
  5. Frequency Analysis
  6. Preliminaries
  7. Problem Formulation
  8. Frequency Representation
  9. Phase Compensation
  10. Mean Normalization
  11. Method
  12. Overview
  13. Normalization
  14. Frequency-Domain Normalization
  15. Time-Domain Normalization
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Comparison of DDN and TimeAPN in the time and frequency domains. In the time domain (left), TimeAPN achieves a lower prediction error (MSE = 0.075) than DDN (MSE = 0.123). In the frequency domain (middle and right), DDN fails to accurately capture the ground-truth amplitude and phase spectra, whereas TimeAPN produces spectra that align more closely with the true amplitude and phase distributions.
  • Figure 2: Overview of the proposed TimeAPN framework. The model first normalizes the input sequence to extract stationary components and non-stationary factors. Stationary sequences are fed into a forecasting model (FM), while non-stationary factors are predicted using the Mean and Phase Prediction Module (MPPM). Predicted non-stationary factors are then combined with FM outputs through de-normalization to reconstruct the future series.
  • Figure 3: Comparison of reversible normalization methods on a forecasting sample from the PatchTST model trained on the Electricity dataset. (a) PatchTST without normalization, (b) PatchTST with RevIN, (c) PatchTST with DDN, and (d) PatchTST with the proposed TimeAPN.