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GCGNet: Graph-Consistent Generative Network for Time Series Forecasting with Exogenous Variables

Zhengyu Li, Xiangfei Qiu, Yuhan Zhu, Xingjian Wu, Jilin Hu, Chenjuan Guo, Bin Yang

arXiv:2603.08032v2cs.LGcs.AI

TL;DR

Forecasting with exogenous variables requires joint temporal and channel correlation modeling, but existing two-step approaches and noisy observations limit robust learning. GCGNet generates coarse predictions, aligns graph structures, and refines outputs, achieving state-of-the-art performance across 12 real-world datasets.

  • Problem

    Forecasting with exogenous variables requires modeling temporal and channel correlations jointly and robustly because two-step methods and noisy observations limit reliable correlation learning.

  • Method

    GCGNet uses a Variational Generator, Graph Structure Aligner, and Graph Refiner to generate, structurally align, and refine forecasts using graph-represented correlations.

  • Results

    GCGNet achieves state-of-the-art forecasting performance across 12 real-world datasets, including 18 first-place MSE rankings and 20 first-place MAE rankings.

  • Takeaways & Limitations

    Graph-consistent generation provides a robust framework for capturing complex joint temporal and channel correlations in forecasting with exogenous variables.

Abstract

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Exogenous variables offer valuable supplementary information for predicting future endogenous variables. Forecasting with exogenous variables needs to consider both past-to-future dependencies (i.e., temporal correlations) and the influence of exogenous variables on endogenous variables (i.e., channel correlations). This is pivotal when future exogenous variables are available, because they may directly affect the future endogenous variables. Many methods have been proposed for time series forecasting with exogenous variables, focusing on modeling temporal and channel correlations. However, most of them use a two-step strategy, modeling temporal and channel correlations separately, which limits their ability to capture joint correlations across time and channels. Furthermore, in real-world scenarios, time series are frequently affected by various forms of noises, underscoring the critical importance of robustness in such correlations modeling. To address these limitations, we propose GCGNet, a Graph-Consistent Generative Network for time series forecasting with exogenous variables. Specifically, GCGNet first employs a Variational Generator to produce coarse predictions. A Graph Structure Aligner then further guides it by evaluating the consistency between the generated and true correlations, where the correlations are represented as graphs, and are robust to noises. Finally, a Graph Refiner is proposed to refine the predictions to prevent degeneration and improve accuracy. Extensive experiments on 12 real-world datasets demonstrate that GCGNet outperforms state-of-the-art baselines.

1 INTRODUCTION

Forecasting with exogenous variables must capture temporal and channel correlations, yet noise and two-step modeling hinder reliable joint correlation learning. GCGNet addresses these issues with graph-consistent generation and refinement, achieving strong results across real-world datasets.

  • 1 INTRODUCTION: Two-step methods model temporal and channel correlations separately, limiting their ability to capture joint dependencies across time and channels.Figure 2 categorizes temporal-first, channel-first, and joint modeling strategies.
  • 1 INTRODUCTION: Sensor failures, transmission errors, and recording mistakes create noisy observations that can cause conventional models to overfit unreliable correlations.These disturbances motivate robustness in correlation modeling.
  • 1 INTRODUCTION: GCGNet uses a Variational Generator, Graph Structure Aligner, and Graph Refiner to produce coarse predictions, enforce graph consistency, and prevent degeneration.The aligner compares generated and true correlation structures, while refinement improves the initial predictions.
  • 1 INTRODUCTION: GCGNet targets robust joint modeling of temporal and channel correlations for forecasting with historical and future exogenous variables.The framework uses graph-consistent generation rather than treating the two correlation types separately.

2 RELATED WORK

Prior work includes classical statistical forecasters and deep generative models for time-series data. Classical approaches impose strong parametric assumptions, while generative approaches learn distributions or latent structures.

  • 2 RELATED WORK: ARIMAX and SARIMAX incorporate correlations between exogenous and endogenous variables but rely on strong parametric assumptions about data functions and distributions.These assumptions may limit flexibility for complex real-world time series.
  • 2 RELATED WORK: TimeGAN, GT-GAN, and VAE-based methods model realistic sequences or latent representations for time-series generation and related tasks.The cited generative approaches optimize supervised or adversarial objectives, or preserve statistical properties through latent representations.

3 METHODOLOGY

GCGNet forecasts future endogenous variables from historical endogenous and exogenous series plus available future exogenous variables by jointly modeling temporal and channel correlations as robust graph structures. Its Variational Generator produces coarse predictions, the Graph Structure Aligner matches generated and ground-truth correlations, and the Graph Refiner uses the learned graph to produce final predictions.

  • 3 METHODOLOGY: GCGNet takes historical endogenous and exogenous series together with available future exogenous variables to forecast future endogenous variables.The forecasting horizon is F, with N endogenous variables and D exogenous variables.
  • 3 METHODOLOGY: The Variational Generator produces a coarse future-sequence prediction that supplies an initial estimate for subsequent correlation modeling.Historical endogenous and exogenous variables are processed by a VAE, and the generated results are combined with observed inputs to form the aligner input.
  • 3 METHODOLOGY: The Graph Structure Aligner converts patch-wise representations of generated and ground-truth full sequences into graphs and minimizes their structural difference.Patchify creates non-overlapping temporal patches, while the shared Graph VAE produces denoised graph representations; the L1 alignment loss encourages matching joint temporal and channel correlations.
  • 3 METHODOLOGY: The Graph Refiner sparsifies the learned adjacency matrix, applies graph convolutions to generated patch embeddings, and projects the refined features into final predictions.Top-k edge selection retains significant temporal and channel connections before multilayer GCN propagation.
  • 3 METHODOLOGY: GCGNet jointly trains alignment, forecasting, and two variational regularization terms in a total loss.The KL terms regularize the latent spaces of the Variational Generator VAE and Graph VAE.

4 EXPERIMENTS

Experiments evaluate GCGNet across 12 real-world datasets, forecasting settings, missing exogenous variables, and model analyses. GCGNet achieves strong benchmark performance and benefits from joint correlation modeling and generative components.

  • 4.1 MAIN RESULTS: The evaluation compares GCGNet with 10 baselines, including methods that do and do not inherently support future exogenous variables.MSE and MAE are evaluated in short-term and long-term prediction settings.
  • 4.1 MAIN RESULTS: GCGNet achieves 18 first-place MSE rankings and 20 first-place MAE rankings across short- and long-term forecasts on 12 datasets.It outperforms methods using two-step modeling and adapted state-of-the-art forecasters.
  • 4.2 MODEL ANALYSES: Ablation studies examine the Variational Generator, alignment loss, Graph VAE, and Graph Refiner to assess their contributions.The reported design forces informative adjacency matrices into prediction and supports temporal and channel correlation capture.
  • 4.2 MODEL ANALYSES: Without future exogenous variables, GCGNet generates exogenous predictions and keeps the remaining forecasting pipeline unchanged.Baselines receive only historical variables under this setting.
  • 4.2 MODEL ANALYSES: Under partially missing exogenous variables, GCGNet outperforms its version without VAE, while masking experiments test zero and random replacements at multiple ratios.The VAE variant is reported to handle missing or noisy inputs and recover useful structure from incomplete data.
  • 4.2 MODEL ANALYSES: On the NP dataset, GCGNet more closely matches electricity-price ground truth than PatchTST and CrossLinear by jointly modeling temporal and channel correlations.With future exogenous variables included, GCGNet’s performance improves further.

5 CONCLUSION

The conclusion presents GCGNet as a graph-consistent generative network for forecasting with exogenous variables. Its pipeline aligns graph structures to model joint temporal and channel correlations robustly, then refines predictions, achieving state-of-the-art performance on diverse real-world datasets.

  • 5 CONCLUSION: GCGNet generates coarse predictions, enforces structural alignment, and refines outputs to capture joint temporal and channel correlations robustly.The design is intended to prevent degeneration and improve accuracy.

ETHICS STATEMENT

The study uses publicly available datasets without personally identifiable information and involves no human subjects.

  • ETHICS STATEMENT: The experiments use publicly available datasets without personally identifiable information, and no human subjects are involved.

REPRODUCIBILITY STATEMENT

The authors state that experimental results are reproducible and provide model code in an anonymous repository.

  • REPRODUCIBILITY STATEMENT: Model code is released in an anonymous repository to support reproduction of the experimental results.

A.1 DATASETS

The evaluation uses 12 real-world datasets with exogenous variables, spanning five EPF electricity-price datasets and seven datasets collected by DAG.

  • A.1 DATASETS: Table 6 reports dataset statistics and distinguishes exogenous variables from endogenous variables.
  • A.1 DATASETS: The benchmark contains 12 real-world datasets with exogenous variables: five EPF datasets and seven DAG-collected datasets.The EPF datasets are NP, PJM, BE, FR, and DE; the DAG datasets include Energy, Colbún, Rapel, Sdwpfh1, Sdwpfh2, Sdwpfm1, and Sdwpfm2.
  • A.1 DATASETS: Future exogenous variables are approximately known or obtainable with high accuracy in all datasets.
  • A.1 DATASETS: NP uses hourly electricity price as the endogenous variable, with grid load and wind power forecast as exogenous features.

A.2 IMPLEMENTATION DETAILS

The implementation standardizes baseline evaluation and uses an MLP fusion module to incorporate future exogenous variables into traditional forecasting models.

  • A.2 IMPLEMENTATION DETAILS: All baselines use drop-last=False to avoid test-batch-size errors during fair comparison.Experiments use PyTorch in Python 3.8 on an NVIDIA Tesla-A800 GPU.
  • A.2 IMPLEMENTATION DETAILS: Figure 5 visualizes prediction results on the NP dataset.
  • A.2 IMPLEMENTATION DETAILS: The backbone generates a latent representation z from historical endogenous and exogenous variables using the forecasting model parameters.
  • A.2 IMPLEMENTATION DETAILS: The MLP fusion module concatenates a backbone latent representation with future exogenous variables to produce final predictions.The backbone receives historical endogenous and exogenous variables before fusion with future exogenous inputs.
  • A.2 IMPLEMENTATION DETAILS: The fusion module has its own parameters, denoted θ_MLP.

B.1 VISUALIZATION OF RESULTS

Visual results show GCGNet capturing joint temporal and channel correlations, with strong performance both when future exogenous variables are available and when they are absent.

  • B.1 VISUALIZATION OF RESULTS: GCGNet consistently outperforms all baselines on NP visualizations with and without future exogenous variables.The target is electricity price, while wind power and grid load serve as exogenous variables.
  • B.1 VISUALIZATION OF RESULTS: PatchTST mainly reproduces historical endogenous patterns because its channel-independent modeling cannot incorporate exogenous variables.
  • B.1 VISUALIZATION OF RESULTS: With future exogenous variables, DUET, CrossLinear, and TimeXer improve, but their gains remain limited relative to GCGNet.
  • B.1 VISUALIZATION OF RESULTS: An MLP-fused PatchTST can use future exogenous variables through two-step modeling, but its predictions largely follow future grid-load shape and improve only modestly.
  • B.1 VISUALIZATION OF RESULTS: Without future exogenous variables, GCGNet maintains strong performance, indicating robustness and generalizability in that setting.

B.3 CONTROLLED HYPERPARAMETER EXPERIMENT

Controlled experiments show that GCGNet performs best across exogenous-variable availability settings, missingness conditions, and varied look-back windows, while sensitivity studies identify favorable architectural ranges.

  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: GCGNet achieves the best performance with fixed configurations both when future exogenous variables are available and when they are unavailable.Tables 8 and 9 report the two controlled settings.
  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: GCGNet consistently achieves the best performance across zero-replacement and random-replacement masking at 10%, 30%, and 50% missingness.The comparison includes TimeXer, CrossLinear, DUET, and PatchTST.
  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: Patch and VAE latent dimensions between 64 and 256 generally perform favorably, whereas excessive dimensions can add redundancy and overfitting risk.
  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: One or two GCN layers perform better than deeper architectures, which may suffer from over-smoothing.
  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: A 50% sparsity ratio provides a favorable balance between removing uninformative weights and retaining critical connections.
  • B.3 CONTROLLED HYPERPARAMETER EXPERIMENT: Across DE, PJM, Energy, and NP, GCGNet outperforms baselines as look-back windows vary and maintains relatively stable metrics as windows increase.The tested windows are 96, 168, 336, 512, and 720; the reported stability indicates robustness to additional noise from longer histories.
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