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Effective Probabilistic Time Series Forecasting with Fourier Adaptive Noise-Separated Diffusion

Xinyan Wang, Rui Dai, Kaikui Liu, Xiangxiang Chu

TL;DR

FALDA tackles long-horizon probabilistic time series forecasting by decomposing targets into non-stationary, stationary, and noise components using Fourier analysis, enabling component-specific modeling. It combines a non-stationary adapter, a flexible time-series backbone, and a conditional diffusion denoiser (DEMA) to learn the aleatoric noise while reducing epistemic uncertainty, with final predictions formed as $\hat{Y}_{\text{FALDA}} = \hat{Y}_{\text{non}} + \hat{Y}_{\text{stat}} + \hat{R}$. The framework is grounded in the Diffusion Model for Residual Regression (DMRR), unifying CARD and standard DDPM dynamics for residual learning and proving equivalence of residual diffusion processes; this enables efficient, non-autoregressive forecasting via DDIM. Empirically, FALDA yields superior point estimates and probabilistic forecasts across six real-world datasets, alongside substantial computational speedups over prior diffusion-based TSF methods, highlighting its practical impact for scalable, accurate forecasting under uncertainty.

Abstract

We propose the Fourier Adaptive Lite Diffusion Architecture (FALDA), a novel probabilistic framework for time series forecasting. First, we introduce the Diffusion Model for Residual Regression (DMRR) framework, which unifies diffusion-based probabilistic regression methods. Within this framework, FALDA leverages Fourier-based decomposition to incorporate a component-specific architecture, enabling tailored modeling of individual temporal components. A conditional diffusion model is utilized to estimate the future noise term, while our proposed lightweight denoiser, DEMA (Decomposition MLP with AdaLN), conditions on the historical noise term to enhance denoising performance. Through mathematical analysis and empirical validation, we demonstrate that FALDA effectively reduces epistemic uncertainty, allowing probabilistic learning to primarily focus on aleatoric uncertainty. Experiments on six real-world benchmarks demonstrate that FALDA consistently outperforms existing probabilistic forecasting approaches across most datasets for long-term time series forecasting while achieving enhanced computational efficiency without compromising accuracy. Notably, FALDA also achieves superior overall performance compared to state-of-the-art (SOTA) point forecasting approaches, with improvements of up to 9%.

Effective Probabilistic Time Series Forecasting with Fourier Adaptive Noise-Separated Diffusion

TL;DR

FALDA tackles long-horizon probabilistic time series forecasting by decomposing targets into non-stationary, stationary, and noise components using Fourier analysis, enabling component-specific modeling. It combines a non-stationary adapter, a flexible time-series backbone, and a conditional diffusion denoiser (DEMA) to learn the aleatoric noise while reducing epistemic uncertainty, with final predictions formed as . The framework is grounded in the Diffusion Model for Residual Regression (DMRR), unifying CARD and standard DDPM dynamics for residual learning and proving equivalence of residual diffusion processes; this enables efficient, non-autoregressive forecasting via DDIM. Empirically, FALDA yields superior point estimates and probabilistic forecasts across six real-world datasets, alongside substantial computational speedups over prior diffusion-based TSF methods, highlighting its practical impact for scalable, accurate forecasting under uncertainty.

Abstract

We propose the Fourier Adaptive Lite Diffusion Architecture (FALDA), a novel probabilistic framework for time series forecasting. First, we introduce the Diffusion Model for Residual Regression (DMRR) framework, which unifies diffusion-based probabilistic regression methods. Within this framework, FALDA leverages Fourier-based decomposition to incorporate a component-specific architecture, enabling tailored modeling of individual temporal components. A conditional diffusion model is utilized to estimate the future noise term, while our proposed lightweight denoiser, DEMA (Decomposition MLP with AdaLN), conditions on the historical noise term to enhance denoising performance. Through mathematical analysis and empirical validation, we demonstrate that FALDA effectively reduces epistemic uncertainty, allowing probabilistic learning to primarily focus on aleatoric uncertainty. Experiments on six real-world benchmarks demonstrate that FALDA consistently outperforms existing probabilistic forecasting approaches across most datasets for long-term time series forecasting while achieving enhanced computational efficiency without compromising accuracy. Notably, FALDA also achieves superior overall performance compared to state-of-the-art (SOTA) point forecasting approaches, with improvements of up to 9%.
Paper Structure (46 sections, 1 theorem, 43 equations, 15 figures, 13 tables, 2 algorithms)

This paper contains 46 sections, 1 theorem, 43 equations, 15 figures, 13 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Diffusion Model for Residual Regression (DMRR)
  3. CARD
  4. The unified framework
  5. Fourier Adaptive Lite Diffusion Architecture (FALDA)
  6. Problem Statement
  7. Main framework
  8. FALDA
  9. DEMA
  10. Analysis of Different Diffusion-based Time Series Models with Residual Learning
  11. Experiments
  12. Experiment Setup
  13. Main Result
  14. Forecasting performance and computational efficiency
  15. Plug-and-play performance
  16. ...and 31 more sections

Key Result

Proposition 1

Let $y_k$ be the Markov chain defined in Eq. eq:card_forward. Let $l_k = y_k - f_{\phi}(x)$, we have: and Thus, the residual process $l_t$ exhibits identical Markovian dynamics to the standard DDPM framework in both forward and reverse processes as shown in Eq. eq:ddpm: multi-step and Eq. eq:ddpm_posterior.

Figures (15)

  • Figure 1: Performance of FALDA in point estimation (MAE, left) and probabilistic prediction (CRPS, right). All three plug-and-play methods (TMDM, D$^3$U, and FALDA) utilize NSformer as the same backbone network for fair comparison.
  • Figure 2: Comparison of three diffusion frameworks: DDPM, CARD, and DMRR, where $\hat{y}_{\text{DDPM}}$, $\hat{y}_{\text{CARD}}$, and $\hat{y}_{\text{DMRR}}$ represent their respective final estimates.
  • Figure 3: An illustration of the proposed FALDA framework. By leveraging Fourier decomposition, NS-Adapter and TS-Backbone generate the preliminary estimation, $\hat{Y}$. The prediction residual $R=Y-\hat{Y}$ is then input into the denoiser for subsequent probabilistic learning and refinement of the preliminary estimation.
  • Figure 4: Evaluation of different training strategies on the ILI Dataset. The left subplot shows the MSE performance, while the right subplot shows the MAE performance. $k'$-DS: fine-tuning with diffusion step $k'$. No-FT: no fine-tuning.
  • Figure 5: Training speed comparison between FALDA and TMDM on the Exchange dataset. The curves depict the evolution of metrics: MSE (left) and MAE (right) across training epochs.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof