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Decorrelating the Future: Joint Frequency Domain Learning for Spatio-temporal Forecasting

Zepu Wang, Bowen Liao, Jeff, Ban

TL;DR

FreST Loss is model-agnostic and consistently improves state-of-the-art baselines by better capturing holistic spatio-temporal dynamics, and reduces estimation bias associated with time-domain training objectives.

Abstract

Standard direct forecasting models typically rely on point-wise objectives such as Mean Squared Error, which fail to capture the complex spatio-temporal dependencies inherent in graph-structured signals. While recent frequency-domain approaches such as FreDF mitigate temporal autocorrelation, they often overlook spatial and cross spatio-temporal interactions. To address this limitation, we propose FreST Loss, a frequency-enhanced spatio-temporal training objective that extends supervision to the joint spatio-temporal spectrum. By leveraging the Joint Fourier Transform (JFT), FreST Loss aligns model predictions with ground truth in a unified spectral domain, effectively decorrelating complex dependencies across both space and time. Theoretical analysis shows that this formulation reduces estimation bias associated with time-domain training objectives. Extensive experiments on six real-world datasets demonstrate that FreST Loss is model-agnostic and consistently improves state-of-the-art baselines by better capturing holistic spatio-temporal dynamics.

Decorrelating the Future: Joint Frequency Domain Learning for Spatio-temporal Forecasting

TL;DR

FreST Loss is model-agnostic and consistently improves state-of-the-art baselines by better capturing holistic spatio-temporal dynamics, and reduces estimation bias associated with time-domain training objectives.

Abstract

Standard direct forecasting models typically rely on point-wise objectives such as Mean Squared Error, which fail to capture the complex spatio-temporal dependencies inherent in graph-structured signals. While recent frequency-domain approaches such as FreDF mitigate temporal autocorrelation, they often overlook spatial and cross spatio-temporal interactions. To address this limitation, we propose FreST Loss, a frequency-enhanced spatio-temporal training objective that extends supervision to the joint spatio-temporal spectrum. By leveraging the Joint Fourier Transform (JFT), FreST Loss aligns model predictions with ground truth in a unified spectral domain, effectively decorrelating complex dependencies across both space and time. Theoretical analysis shows that this formulation reduces estimation bias associated with time-domain training objectives. Extensive experiments on six real-world datasets demonstrate that FreST Loss is model-agnostic and consistently improves state-of-the-art baselines by better capturing holistic spatio-temporal dynamics.
Paper Structure (32 sections, 3 theorems, 15 equations, 5 figures, 4 tables)

This paper contains 32 sections, 3 theorems, 15 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Spatio-temporal Forecasting
  4. Frequency Domain Methods in Forecasting
  5. Problem Formulation and Motivation
  6. Spatio-Temporal Forecasting
  7. Spatio-Temporal Signals.
  8. Direct Forecasting Paradigm.
  9. Limitation of DF and Motivation of FreST Loss
  10. Optimization Objective and Limitations.
  11. Discussion on Correlations in the Frequency Domain
  12. Decorrelating the Temporal Dimension
  13. Decorrelating the Spatial Dimension
  14. Decorrelating the Cross-Spatio-Temporal Dimension
  15. Empirical Analysis
  16. ...and 17 more sections

Key Result

theorem 1

The learning objective of the standard DF paradigm (Eq. eq:mse_loss) is biased with respect to the true negative log-likelihood (NLL) of the data distribution. By generalizing the analysis to both spatial and temporal dimensions, this bias can be explicitly decomposed as: where $\sigma$ denotes the standard deviation of the intrinsic noise (assumed to be constant). The term $\mu_{i,n|\text{pa}}$

Figures (5)

  • Figure 1: Correlation analysis of spatio-temporal data under different spectral transforms.
  • Figure 2: Overall Framework of FreST Loss.
  • Figure 3: Performance comparison of STGCN in AIR-BJ and iTransformer in Air-GZ with various future sequence lengths
  • Figure 4: Training and Validation RMSE Curve for Staeformer in SH-METRO
  • Figure 5: Impact of $\alpha$ on Model Performance in AIR-BJ

Theorems & Definitions (6)

  • theorem 1: Bias of DF in the Spatio-Temporal Domain
  • definition 1: Fast Fourier Transform
  • theorem 2: Temporal Decorrelation via FFT
  • definition 2: Graph Fourier Transform
  • theorem 3: Spatial Decorrelation via GFT
  • definition 3: Joint Spatio-Temporal Fourier Transform