A Decomposable Forward Process in Diffusion Models for Time-Series Forecasting

TL;DR

This work addresses loss of structured temporal information in diffusion-based time-series forecasting by introducing a decomposable forward diffusion process that performs stage-wise noise diffusion across spectral components identified by Fourier or Wavelet decompositions. The method is model-agnostic and compatible with existing backbones, and it preserves dominant seasonal and trend structures longer in the forward pass, improving long-horizon forecast quality with minimal computational overhead. The authors provide a general theoretical framework for component-wise diffusion, derive closed-form forward steps, and demonstrate empirical improvements across diverse real-world datasets, particularly those with strong seasonality, while offering insights into component selection and scheduler design. The approach enhances interpretability by mapping each diffusion stage to an interpretable signal component, enabling more reliable and explainable long-range predictions in applications such as energy, health, and finance.

Abstract

We introduce a model-agnostic forward diffusion process for time-series forecasting that decomposes signals into spectral components, preserving structured temporal patterns such as seasonality more effectively than standard diffusion. Unlike prior work that modifies the network architecture or diffuses directly in the frequency domain, our proposed method alters only the diffusion process itself, making it compatible with existing diffusion backbones (e.g., DiffWave, TimeGrad, CSDI). By staging noise injection according to component energy, it maintains high signal-to-noise ratios for dominant frequencies throughout the diffusion trajectory, thereby improving the recoverability of long-term patterns. This strategy enables the model to maintain the signal structure for a longer period in the forward process, leading to improved forecast quality. Across standard forecasting benchmarks, we show that applying spectral decomposition strategies, such as the Fourier or Wavelet transform, consistently improves upon diffusion models using the baseline forward process, with negligible computational overhead. The code for this paper is available at https://anonymous.4open.science/r/D-FDP-4A29.

Figures (7)

• Figure 1: Diffusion process with decomposition. During the forward process, noise is added to each component in ascending order of amplitude. This process slows down the destruction of the more relevant frequencies, preserving key structures longer.
• Figure 2: Effect of SNR scaling in noise application. Key frequencies in the signal remain discernible for a longer duration. Without SNR scaling, noise spreads across components, degrading separability. All approaches in this figure use a linear schedule $0.002 < \beta < 0.02$.
• Figure 3: forecasting comparison for a 300 steps window length, of choosing a different number of components.
• Figure 4: Evolution of the noise scheduler with different number of components. As the number of components increase, the smoothness of the diffusion process decreases, for fixed scheduler parameters.
• Figure 5: Exploration of the feasibility region for $T \in [30,200], \beta_\tau \in [0.01,0.3], \beta_0 \in [0.0001, 0.05]$. The gray area indicates where $b_\tau > b_0$,with the green area indicating the feasibility space.