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A Unified Frequency Domain Decomposition Framework for Interpretable and Robust Time Series Forecasting

Cheng He, Xijie Liang, Zengrong Zheng, Patrick P. C. Lee, Xu Huang, Zhaoyi Li, Hong Xie, Defu Lian, Enhong Chen

TL;DR

The paper tackles the challenge of interpretable and robust time-series forecasting under concept drift and basis evolution. It introduces FIRE, a unified frequency-domain decomposition framework that independently models amplitude and phase, learns adaptive frequency-basis weights via causal attention, and optimizes with a composite loss designed to address basis evolution and drift. Empirical evaluations across diverse long-horizon datasets show that FIRE achieves state-of-the-art accuracy while enhancing interpretability, outperforming both purely time-domain and prior frequency-domain methods. The approach provides a principled, mathematically grounded path for robust forecasting in dynamic environments, with practical implications for industrial time-series applications.

Abstract

Current approaches for time series forecasting, whether in the time or frequency domain, predominantly use deep learning models based on linear layers or transformers. They often encode time series data in a black-box manner and rely on trial-and-error optimization solely based on forecasting performance, leading to limited interpretability and theoretical understanding. Furthermore, the dynamics in data distribution over time and frequency domains pose a critical challenge to accurate forecasting. We propose FIRE, a unified frequency domain decomposition framework that provides a mathematical abstraction for diverse types of time series, so as to achieve interpretable and robust time series forecasting. FIRE introduces several key innovations: (i) independent modeling of amplitude and phase components, (ii) adaptive learning of weights of frequency basis components, (iii) a targeted loss function, and (iv) a novel training paradigm for sparse data. Extensive experiments demonstrate that FIRE consistently outperforms state-of-the-art models on long-term forecasting benchmarks, achieving superior predictive performance and significantly enhancing interpretability of time series

A Unified Frequency Domain Decomposition Framework for Interpretable and Robust Time Series Forecasting

TL;DR

The paper tackles the challenge of interpretable and robust time-series forecasting under concept drift and basis evolution. It introduces FIRE, a unified frequency-domain decomposition framework that independently models amplitude and phase, learns adaptive frequency-basis weights via causal attention, and optimizes with a composite loss designed to address basis evolution and drift. Empirical evaluations across diverse long-horizon datasets show that FIRE achieves state-of-the-art accuracy while enhancing interpretability, outperforming both purely time-domain and prior frequency-domain methods. The approach provides a principled, mathematically grounded path for robust forecasting in dynamic environments, with practical implications for industrial time-series applications.

Abstract

Current approaches for time series forecasting, whether in the time or frequency domain, predominantly use deep learning models based on linear layers or transformers. They often encode time series data in a black-box manner and rely on trial-and-error optimization solely based on forecasting performance, leading to limited interpretability and theoretical understanding. Furthermore, the dynamics in data distribution over time and frequency domains pose a critical challenge to accurate forecasting. We propose FIRE, a unified frequency domain decomposition framework that provides a mathematical abstraction for diverse types of time series, so as to achieve interpretable and robust time series forecasting. FIRE introduces several key innovations: (i) independent modeling of amplitude and phase components, (ii) adaptive learning of weights of frequency basis components, (iii) a targeted loss function, and (iv) a novel training paradigm for sparse data. Extensive experiments demonstrate that FIRE consistently outperforms state-of-the-art models on long-term forecasting benchmarks, achieving superior predictive performance and significantly enhancing interpretability of time series

Paper Structure

This paper contains 27 sections, 1 theorem, 28 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytical Formulation of Time Series Data
  4. Concept Drift in Time Domain
  5. Basis Evolution in Frequency Domain
  6. Strong and Weak Convergence
  7. FIRE Design
  8. Model Architecture
  9. Embedding and Transformation
  10. Frequency Domain Backbone
  11. Output Projection
  12. Loss Function
  13. Discussion
  14. Experiments
  15. Datasets and Baselines
  16. ...and 12 more sections

Key Result

Lemma 3.1

A non-stationary time series with time-varying distribution exhibits concept drift. Under linear time-invariant signal decomposition, modeling distributional shifts in the temporal domain is equivalent to modeling independent changes in amplitude and phase in the frequency domain.

Figures (4)

  • Figure 1: Model architecture of FIRE. It tranforms multivariate time series into the frequency domain through a sequence of steps including CI, IN, patching, and FFT. It captures intra-patch correlations using complex linear layers. It models concept drift via linear transformations, and basis evolution via causal attention mechanisms. It finally generates predictions by a flattened linear projection layer.
  • Figure 2: Variations in amplitude and phase distributions between consecutive frequency patches in the frequency domain. The patches are sampled from the Weather and Etth1 datasets, respectively.
  • Figure 3: Model scalability analysis
  • Figure 4: Average forecasting results on ETTh1 and Weather datasets with various look back window lengths.

Theorems & Definitions (5)

  • Definition 1: Degree of concept drift
  • Definition 2: Basis evolution criterion
  • Definition 3: Patch-level basis evolution
  • Definition 4: Degree of basis evolution
  • Lemma 3.1: Equivalence of Concept Drift Modeling in Temporal and Frequency Domains