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Series-to-Series Diffusion Bridge Model

Hao Yang, Zhanbo Feng, Feng Zhou, Robert C Qiu, Zenan Ling

TL;DR

This paper revisits time series diffusion models and proposes a novel diffusion-based time series forecasting model, the Series-to-Series Diffusion Bridge Model, which leverages the Brownian Bridge process to reduce randomness in reverse estimations and improves accuracy by incorporating informative priors and conditions derived from historical time series data.

Abstract

Diffusion models have risen to prominence in time series forecasting, showcasing their robust capability to model complex data distributions. However, their effectiveness in deterministic predictions is often constrained by instability arising from their inherent stochasticity. In this paper, we revisit time series diffusion models and present a comprehensive framework that encompasses most existing diffusion-based methods. Building on this theoretical foundation, we propose a novel diffusion-based time series forecasting model, the Series-to-Series Diffusion Bridge Model ($\mathrm{S^2DBM}$), which leverages the Brownian Bridge process to reduce randomness in reverse estimations and improves accuracy by incorporating informative priors and conditions derived from historical time series data. Experimental results demonstrate that $\mathrm{S^2DBM}$ delivers superior performance in point-to-point forecasting and competes effectively with other diffusion-based models in probabilistic forecasting.

Series-to-Series Diffusion Bridge Model

TL;DR

This paper revisits time series diffusion models and proposes a novel diffusion-based time series forecasting model, the Series-to-Series Diffusion Bridge Model, which leverages the Brownian Bridge process to reduce randomness in reverse estimations and improves accuracy by incorporating informative priors and conditions derived from historical time series data.

Abstract

Diffusion models have risen to prominence in time series forecasting, showcasing their robust capability to model complex data distributions. However, their effectiveness in deterministic predictions is often constrained by instability arising from their inherent stochasticity. In this paper, we revisit time series diffusion models and present a comprehensive framework that encompasses most existing diffusion-based methods. Building on this theoretical foundation, we propose a novel diffusion-based time series forecasting model, the Series-to-Series Diffusion Bridge Model (), which leverages the Brownian Bridge process to reduce randomness in reverse estimations and improves accuracy by incorporating informative priors and conditions derived from historical time series data. Experimental results demonstrate that delivers superior performance in point-to-point forecasting and competes effectively with other diffusion-based models in probabilistic forecasting.

Paper Structure

This paper contains 27 sections, 2 theorems, 26 equations, 8 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Diffusion-based Time Series Forecasting.
  4. Diffusion Bridge.
  5. Methodology
  6. Preliminaries
  7. Revisiting Generalized Diffusion Model for Time Series
  8. Series-to-Series Diffusion Bridge Model
  9. Linear Model based Conditioning Method.
  10. Label-Guided Data Estimation.
  11. Experiments
  12. Experimental Settings
  13. Datasets.
  14. Baselines.
  15. Evaluation metrics.
  16. ...and 12 more sections

Key Result

Theorem 1

The non-autoregressive diffusion processes in time series can be formalized as follows: The reverse diffusion process corresponding to $\hat{\beta}_t \neq 0$ can be formulated as: where $\hat{\alpha}_t$, $\hat{\beta}_t$, and $\hat{\gamma}t$ are time-dependent scaling factors. The vector ${\bm{h}} = F({\bm{x}})$ acts as the conditional representation incorporating prior knowledge, with $F(\cdot)$

Figures (8)

  • Figure 1: Examples of time series forecasting for the ETTh1 dataset. The length of forecast windows is 96. The purple line shows the ground truth. For CSDI and TMDM, median values of probabilistic forecasting are shown as the line and 5% and 95% quantiles are shown as the shade. The point-to-point forecasting results of our $\mathrm{S^2DBM}$ are shown as the orange line.
  • Figure 2: An illustration of the proposed $\mathrm{S^2DBM}$
  • Figure 3: Visualizations on ETTh1 by CSDI, TMDM and the proposed $\mathrm{S^2DBM}$.
  • Figure 4: Visualizations on ETTh1 by Conditional DDPM and the proposed $\mathrm{S^2DBM}$.
  • Figure 5: The predicted samples by our $\mathrm{S^2DBM}$ model for different forecast window lengths on the ETTh1 dataset.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1: Brownian Bridge between Historical and Predicted Time Series
  • Remark 1: The reverse process of $\mathrm{S^2DBM}$
  • Example 1: Point-to-point forecasting
  • Example 2: Probabilistic forecasting