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FreqLens: Interpretable Frequency Attribution for Time Series Forecasting

Chi-Sheng Chen, Xinyu Zhang, En-Jui Kuo, Guan-Ying Chen, Qiuzhe Xie, Fan Zhang

TL;DR

FreqLens tackles the interpretability gap in time-series forecasting by jointly learning frequency bases from data and providing axiomatic, frequency-level attribution. It introduces a sigmoid-parametrized frequency decomposition with diversity regularization, a differentiable sparse selection mechanism, and an axiomatic attribution head that yields per-frequency contributions equivalent to Shapley values under an additive decomposition. Across seven benchmarks, including Weather and Traffic, FreqLens achieves competitive or superior forecasting performance while discovering physically meaningful frequencies such as daily and 12-hour cycles, even with limited input windows. Extensive ablations and significance tests support the robustness of both its predictive and interpretability claims, illustrating genuine frequency-level knowledge discovery with formal attribution guarantees. The work enables domain-relevant, interpretable forecasting by linking predictions to tangible periodic patterns without requiring domain-specific priors.

Abstract

Time series forecasting models often lack interpretability, limiting their adoption in domains requiring explainable predictions. We propose \textsc{FreqLens}, an interpretable forecasting framework that discovers and attributes predictions to learnable frequency components. \textsc{FreqLens} introduces two key innovations: (1) \emph{learnable frequency discovery} -- frequency bases are parameterized via sigmoid mapping and learned from data with diversity regularization, enabling automatic discovery of dominant periodic patterns without domain knowledge; and (2) \emph{axiomatic frequency attribution} -- a theoretically grounded framework that provably satisfies Completeness, Faithfulness, Null-Frequency, and Symmetry axioms, with per-frequency attributions equivalent to Shapley values. On Traffic and Weather datasets, \textsc{FreqLens} achieves competitive or superior performance while discovering physically meaningful frequencies: all 5 independent runs discover the 24-hour daily cycle ($24.6 \pm 0.1$h, 2.5\% error) and 12-hour half-daily cycle ($11.8 \pm 0.1$h, 1.6\% error) on Traffic, and weekly cycles ($10\times$ longer than the input window) on Weather. These results demonstrate genuine frequency-level knowledge discovery with formal theoretical guarantees on attribution quality.

FreqLens: Interpretable Frequency Attribution for Time Series Forecasting

TL;DR

FreqLens tackles the interpretability gap in time-series forecasting by jointly learning frequency bases from data and providing axiomatic, frequency-level attribution. It introduces a sigmoid-parametrized frequency decomposition with diversity regularization, a differentiable sparse selection mechanism, and an axiomatic attribution head that yields per-frequency contributions equivalent to Shapley values under an additive decomposition. Across seven benchmarks, including Weather and Traffic, FreqLens achieves competitive or superior forecasting performance while discovering physically meaningful frequencies such as daily and 12-hour cycles, even with limited input windows. Extensive ablations and significance tests support the robustness of both its predictive and interpretability claims, illustrating genuine frequency-level knowledge discovery with formal attribution guarantees. The work enables domain-relevant, interpretable forecasting by linking predictions to tangible periodic patterns without requiring domain-specific priors.

Abstract

Time series forecasting models often lack interpretability, limiting their adoption in domains requiring explainable predictions. We propose \textsc{FreqLens}, an interpretable forecasting framework that discovers and attributes predictions to learnable frequency components. \textsc{FreqLens} introduces two key innovations: (1) \emph{learnable frequency discovery} -- frequency bases are parameterized via sigmoid mapping and learned from data with diversity regularization, enabling automatic discovery of dominant periodic patterns without domain knowledge; and (2) \emph{axiomatic frequency attribution} -- a theoretically grounded framework that provably satisfies Completeness, Faithfulness, Null-Frequency, and Symmetry axioms, with per-frequency attributions equivalent to Shapley values. On Traffic and Weather datasets, \textsc{FreqLens} achieves competitive or superior performance while discovering physically meaningful frequencies: all 5 independent runs discover the 24-hour daily cycle (h, 2.5\% error) and 12-hour half-daily cycle (h, 1.6\% error) on Traffic, and weekly cycles ( longer than the input window) on Weather. These results demonstrate genuine frequency-level knowledge discovery with formal theoretical guarantees on attribution quality.
Paper Structure (46 sections, 1 theorem, 15 equations, 10 figures, 14 tables, 1 algorithm)

This paper contains 46 sections, 1 theorem, 15 equations, 10 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Time Series Forecasting
  4. Interpretability for Time Series
  5. Positioning of FreqLens
  6. Method
  7. Overview
  8. Learnable Frequency Decomposition
  9. Sparse Frequency Selection
  10. Axiomatic Attribution Head
  11. Training Objective
  12. Post-hoc Frequency Interpretation
  13. Experiments
  14. Experimental Setup
  15. Performance Results
  16. ...and 31 more sections

Key Result

theorem 1

Under the additive decomposition eq:additive, the attribution $\mathcal{A}(f) = \text{contribution}(f)$ is the unique function satisfying axioms A1--A4. Moreover, it equals the Shapley value of the cooperative game $(S, v)$ where $v(\mathcal{T}) = M(\mathcal{T}) = \sum_{f \in \mathcal{T}} \text{cont

Figures (10)

  • Figure 1: FreqLens architecture. Input is projected and decomposed into learnable frequency bases, from which top-$K$ are selected. Each selected frequency independently produces a contribution; their sum forms the frequency prediction. A residual path handles non-periodic components.
  • Figure 2: Forecasting comparison on ETTh1 dataset ($H=96$). FreqLens predictions (purple) closely match ground truth (blue), outperforming iTransformer (orange) and Autoformer (green). The model captures both periodic patterns and non-periodic trends.
  • Figure 3: Frequency discovery on Traffic ($H{=}96$, seed=42). Each bar represents one of 32 learned frequency bases, sorted by period. Green bars indicate frequencies matching known physical periods ($<$20% error). Dashed reference lines mark half-daily (12h), daily (24h), weekly (168h), monthly (720h), and yearly (8760h) cycles.
  • Figure 4: Frequency discovery on Weather ($H{=}96$, seed=42). Despite the input window spanning only 16 hours, FreqLens discovers the weekly cycle (160h, 4.8% error) --- $10\times$ longer than the input. The daily and half-daily cycles are also captured.
  • Figure 5: Frequency discovery visualization on Weather dataset ($H=192$, seed=2024). Learned frequencies (bars) align with known physical periods (dashed lines): daily (24h), half-daily (12h), and weekly (168h) cycles.
  • ...and 5 more figures

Theorems & Definitions (2)

  • definition 1: Frequency Attribution Axioms
  • theorem 1: Shapley Equivalence