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Generative Time Series Forecasting with Diffusion, Denoise, and Disentanglement

Yan Li, Xinjiang Lu, Yaqing Wang, Dejing Dou

arXiv:2301.03028v1cs.LG

TL;DR

Real-world time series are often short and noisy, creating a gap for deep forecasting models. D3VAE combines a bidirectional VAE with coupled diffusion, denoising score matching, and latent disentanglement; experiments report state-of-the-art performance and substantial reductions in forecasting errors.

  • Problem

    Short, noisy real-world time series create overfitting and generalization issues for forecasting models.

  • Method

    D3VAE combines a bidirectional VAE backbone with coupled diffusion, denoising score matching, and disentangled multivariate latent variables.

  • Results

    D3VAE achieves consistent state-of-the-art performance on real-world datasets except Wind at prediction length 16, with averaged 43% MSE and 23% CRPS reductions.

  • Takeaways & Limitations

    Experiments on synthetic and real-world data validate D3VAE's state-of-the-art performance against competitive generative models.

  • Takeaways & Limitations

    Diffusion steps and the variance schedule must be chosen cautiously because the coupled process introduces bias that can propagate through the VAE output.

Abstract

from arXiv · show

Time series forecasting has been a widely explored task of great importance in many applications. However, it is common that real-world time series data are recorded in a short time period, which results in a big gap between the deep model and the limited and noisy time series. In this work, we propose to address the time series forecasting problem with generative modeling and propose a bidirectional variational auto-encoder (BVAE) equipped with diffusion, denoise, and disentanglement, namely D3VAE. Specifically, a coupled diffusion probabilistic model is proposed to augment the time series data without increasing the aleatoric uncertainty and implement a more tractable inference process with BVAE. To ensure the generated series move toward the true target, we further propose to adapt and integrate the multiscale denoising score matching into the diffusion process for time series forecasting. In addition, to enhance the interpretability and stability of the prediction, we treat the latent variable in a multivariate manner and disentangle them on top of minimizing total correlation. Extensive experiments on synthetic and real-world data show that D3VAE outperforms competitive algorithms with remarkable margins. Our implementation is available at https://github.com/PaddlePaddle/PaddleSpatial/tree/main/research/D3VAE.

1 Introduction

Neural forecasting methods capture temporal patterns but remain challenged by uncertainty, while VAEs offer latent-distribution modeling and interpretability that can be weakened by entangled variables. D3VAE addresses noisy, short real-world series with coupled diffusion, denoising, and disentanglement.

  • Neural forecasting methods use recurrent, convolutional, or attention mechanisms to model temporal dependencies and complex time-series patterns.
  • Uncertainty remains a major issue for neural-network time-series forecasting, while VAR-based distribution modeling offers reliability but unsatisfactory performance.
  • VAEs model latent data distributions and can improve forecasting interpretability, but entangled latent variables may reduce that interpretability.
  • Real-world series are often noisy and short, creating overfitting and generalization issues that motivate a generative forecasting approach.
  • D3VAE combines coupled diffusion, multiscale denoising score matching, and latent disentanglement to improve generalization, generated-result accuracy, and interpretability.

2 Methodology

D3VAE combines coupled diffusion, BVAE inference, multiscale denoising, and latent disentanglement to forecast short, noisy time series. The framework augments inputs and targets together, guides predictions toward clean targets, and uses disentangled latent states for interpretability and reliability.

  • Generative Time Series Forecasting: D3VAE models forecasting by learning expressive latent representations from input series and sampling target series from the learned generative distribution.The latent representation is intended to capture useful signals from X while mapping the input into a high-expressiveness latent space.
  • Coupled Diffusion Process: The coupled diffusion process simultaneously augments input and target series with separate variance schedules, expanding the distribution space for short time series.The input uses β, while the target uses β′; the process is designed to reduce uncertainty from both generation and inherent data noise.
  • Bidirectional Variational Auto-Encoder: BVAE replaces the conventional diffusion reverse process to provide tractable, supervised forecasting inference from diffused inputs.Its latent states are organized multivariately, with successive states conditioned on earlier states and the input series.
  • Scaled Denoising Score Matching: Multiscale denoising score matching cleans generated targets by using noise levels tied to the diffusion schedule and a gradient denoising jump at inference.This addresses the tendency of generated distributions to move toward corrupted diffused targets; the estimated uncertainty term represents noise between generated and cleaned targets.
  • Disentangling Latent Variables: D3VAE disentangles multivariate latent variables by minimizing averaged total correlation while preserving useful semantics through BVAE’s bidirectional structure.Lower total correlation generally indicates better disentanglement when latent variables retain meaningful information, while excessively low correlation can also reflect uninformative latents.

3 Experiments

Experiments evaluate D3VAE on synthetic and real-world datasets using MSE and CRPS, with additional studies examining uncertainty, disentanglement, diffusion components, and diffusion settings.

  • Toy Datasets: D3VAE achieves state-of-the-art performance most of the time on the two synthetic datasets, with competitive CRPS on D2 at prediction length 16.VAEs outperform VARs and GP on D1, whereas VARs perform better on D2.
  • Real-World Datasets: D3VAE achieves consistent state-of-the-art performance on real-world datasets except Wind at prediction length 16.Under input-8-predict-8, MSE reductions are 90% for Traffic, 71% for Electricity, 48% for Wind, 43% for ETTm1, 40% for ETTh1, and 28% for Weather.
  • Real-World Datasets: 43% averaged MSE reduction and 23% averaged CRPS reduction are achieved across the reported real-world settings.These averages summarize the above input-8-predict-8 and input-16-predict-16 comparisons.
  • Uncertainty Estimation: Estimated uncertainty grows rapidly when extreme values are encountered in Traffic predictions.The colored envelope represents the estimated uncertainty, and the last six dimensions are treated as target variables.
  • Disentanglement and Ablation Studies: Experiments assess disentanglement by classifying latent-variable dimensions and also evaluate the coupled diffusion model through variants that remove diffusion or denoising components.The coupled-diffusion ablation compares D3VAE with variants omitting diffused target series, diffused input series, all diffusion, or denoising score matching.
  • Variance Schedule β and Diffusion Steps: The variance schedule β and diffusion-step count T must be configured properly because insufficient diffusion can be meaningless and excessive diffusion can become uncontrolled.The study varies β_t from 0.01 to 0.1 and T from 100 to 4000 on Electricity.

4 Discussion

The discussion finds that single-step denoising is sufficient for D3VAE target generation, while additional sampling steps increase complexity without improving forecasting. It also identifies diffusion-schedule bias as an application constraint and points to future work on labeled disentanglement factors and sampling methods.

  • Sampling Strategies: Single-step gradient denoising is used to generate the target series, and experiments demonstrate its effectiveness.
  • Sampling Strategies: More sampling steps might not improve prediction performance and instead lead to higher computational complexity.
  • Sampling Strategies: Varying the Langevin noise scale ρ does not bring indispensable benefits for time series forecasting.
  • Limitations: The coupled diffusion model reduces aleatoric uncertainty but introduces bias, so diffusion steps and variance schedules require cautious selection.
  • Limitations: Future work includes labeling time-series factors for broader disentanglement applications and exploring additional generative and sampling methods.

5 Conclusion

The paper concludes that D3VAE combines a bidirectional VAE backbone with coupled diffusion, scaled denoising, and latent disentanglement for generative time-series forecasting. It targets short, noisy data and reports state-of-the-art performance against competitive generative models.

  • Conclusion: D3VAE uses a bidirectional VAE backbone, coupled diffusion, scaled denoising, and latent disentanglement for generative time-series forecasting.
  • Motivation: Short and small real-world time series motivate augmenting input and output series with coupled diffusion without increasing aleatoric uncertainty.
  • Method: A multiscale score-matching denoising network is used to guide generated target series toward accurate future predictions.
  • Related Work: Existing synthetic-series and transformation-based augmentation methods may fail to respect original feature relationships across time or address all limited-data settings.
  • Related Work: Existing uncertainty-estimation work often focuses on epistemic uncertainty, while aleatoric uncertainty from measurement and collection noise is frequently ignored.
  • Related Work: Traditional preprocessing and neural denoising methods have limitations across irregular or diverse time-series types.
  • Related Work: Latent-space representations and disentanglement address interpretability and reliability concerns in multivariate time-series forecasting.

B Proofs of Lemma 1 and Lemma 2

The proofs establish that the coupled diffusion model can capture the ideal target component and reduce the discrepancy between diffusion noise and generation noise. Together, these results support the model’s approximation and noise-matching properties.

  • Lemma 1: Lemma 1 states that a probabilistic model can make the KL divergence between the coupled diffusion target distribution and model distribution arbitrarily small.
  • Lemma 1: The proof argues that the ideal component of the ground-truth target can be fully captured, with reconstruction error tending to zero.
  • Lemma 2: Lemma 2 states that the coupled diffusion process reduces the difference between diffusion noise and generation noise as t approaches infinity.
  • Lemma 2: The proof decomposes target noise into independent estimation and residual components, yielding a factorized noise distribution.
  • Lemma 2: The proof compares KL divergences and uses Gaussian-noise behavior in the limit to establish the noise-reduction result.

C.1 Experimental Settings

The experiments use descriptive statistics from six real-world forecasting datasets and compare forecasting processes across generative baselines. The implementation embeds inputs before a two-layer GRU and configures diffusion schedules by dataset.

  • Datasets: Table 5 reports the main descriptive statistics of the real-world datasets used in the experiments.
  • Input Representation: Inputs are embedded and passed to an RNN to extract temporal dependencies before concatenation.
  • Experimental Settings: The forecasting-process comparison includes DeepAR, TimeGrad, and GP-copula with a sliding step of 1.
  • Input Representation: The input representation uses a two-layer GRU with hidden-state dimensionality 128 and embedding dimensionality 64.
  • Diffusion Configuration: Diffusion schedules use dataset-specific βt ranges and T values of 100 or 1000.

C.2 Implementation Details of Baselines

The experiments compare D3VAE with generative forecasting baselines under short- and long-term settings, full datasets, and varied forecasting conditions. D3VAE is reported to perform strongly across these evaluations, while its sensitivity and scalability are also examined.

  • The baselines include GP-copula, DeepAR, and TimeGrad, representing Gaussian-process, autoregressive recurrent, and diffusion-based probabilistic forecasting approaches.
  • DeepAR, TimeGrad, and GP-copula use chronologically ordered rolling-window samples and revert their CDF outputs to the real distribution for testing.
  • D3VAE consistently outperforms competitive baselines under longer-term forecasting settings evaluated with MSE and CRPS.
  • 52% and 54% MSE reductions, plus 34% and 36% CRPS reductions, are reported for Electricity under input-16-predict-16 and input-32-predict-32 settings, respectively.The comparisons are against the second-best results on the 100%-Electricity dataset.
  • D3VAE provides accurate forecasts of trends and variations on Traffic and Electricity, and its prediction performance depends on the choice of ω.Larger or smaller ω values can produce predictions far from the ground truth.
  • D3VAE’s predictive performance is relatively stable across different series lengths and dataset sizes, except for degraded results on irregular 20%-Electricity data.Longer target series may worsen performance, while reduced data amounts produce more stable results than expected.

D.4 Sensitivity Analysis of Trade-off Parameters in Reconstruction Loss L

The sensitivity analysis evaluates how reconstruction-loss trade-off parameters affect D3VAE’s prediction performance across multiple forecasting horizons.

  • Relative MSE is measured while varying ψ, λ, and γ under input-8-predict-8, input-16-predict-16, and input-32-predict-32 settings.The tested ranges are ψ from 0 to 0.8, λ from 0 to 1.6, and γ from 0 to 0.5.

D.5 Scalability Analysis of Varying Time Series Length and Dataset Size

The scalability analysis tests D3VAE across different time-series lengths and amounts of available data on Electricity and Traffic.

  • D3VAE’s predictive performance remains relatively stable across different series lengths and dataset sizes.Performance may worsen for longer target series, but remains more stable than expected when data are reduced.
  • The 20%-Electricity subset performs much worse because its sliced data contain more irregular values.

E Disentanglement for Time Series Forecasting

D3VAE disentangles multivariate latent variables for time-series forecasting and evaluates the resulting representations with discriminator-based analyses and MIG. The analysis finds that stronger disentanglement weighting improves factor quality, while more factors increase difficulty.

  • Limitation: Choosing suitable disentanglement factors is difficult under unsupervised disentanglement learning.
  • Disentanglement mechanism: D3VAE decomposes each latent variable zi into m factors and minimizes total correlation to reduce interdependencies among hand-crafted factors.The intended result is factors that are both discriminative and informative.
  • Disentanglement evaluation: A discriminator is trained on factor-label instances constructed from different latent dimensions to evaluate disentanglement quality indirectly.Algorithm 3 describes discriminator initialization, factor decomposition, loss accumulation, and optimization until convergence.
  • Disentanglement evaluation: MIG provides a direct measure of disentanglement quality by comparing mutual information between latent factors and generative factors.The supplied formulation uses mutual-information values and the second-largest value in the comparison.
  • Results: Higher γ produces higher-quality disentangled factors on ETTm1 and ETTh1, according to MIG.
  • Results: Disentanglement becomes more difficult as the number of factors increases.
  • Diffusion analysis: Larger diffusion steps or wider variance schedules progressively move diffused series farther from the original data, potentially losing temporal dependencies.

G Necessity of Data Augmentation for Time Series Forecasting

The synthetic experiment demonstrates that limited time-series data can cause overfitting and poor forecasting performance, motivating data augmentation for deep models. An RNN study and diffusion visualizations examine how dataset size and diffusion settings affect the data and losses.

  • Limited time-series data can cause overfitting and poor performance when deep models are used for forecasting.A two-layer RNN is evaluated to demonstrate the number of time points needed for generalization.
  • The synthetic dataset contains d-dimensional observations generated from a recurrent stochastic process and linearly transformed through F.The construction uses wt ∈ R2, F ∈ R2×d, d = 5, Gaussian noise terms, and an input-8-predict-8 window.
  • As the synthetic dataset grows, the inflection points of the RNN training and testing loss curves move back gradually and disappear.The experiment uses an input-8-predict-8 rolling window, a 7:3 train-test split, and at most 100 training epochs.
  • Figure 15 compares original and diffused synthetic series under different variance schedules and diffusion steps.The figure uses a randomly selected sample from synthetic dataset D2.
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