Generative Forecasting Transformer (GFT)

• Generative Forecasting Transformer (GFT) is a hybrid model that merges generative synthesis with transformer-based prediction for improved long-range time series forecasting.
• It leverages a conditional Wasserstein GAN and information theoretic clustering to generate representative synthetic data and balance the bias–variance trade-off.
• Empirical results demonstrate that GFT reduces forecasting error by 5–11% and uses 15–50% fewer parameters compared to state-of-the-art forecasting methods.

The Generative Forecasting Transformer (GFT) is a two-stage hybrid framework for long-range time series forecasting, designed to overcome limitations inherent in both direct and iterative multi-step prediction strategies. GFT achieves this by combining a generative model for synthetic data creation with a transformer-based predictor, supported by an information theoretic clustering approach to enhance sample representativeness. The result is improved bias–variance trade-off, reduced forecasting error, and greater parameter efficiency relative to established benchmarks.

1. Generative Forecasting Strategy

Traditional long-range time series forecasting employs either Direct Forecasting (DF)—which is low bias but high variance as horizon $N$ increases—or Iterative Forecasting (IF), which can reduce variance at the cost of accumulating bias. The generative forecasting strategy (GenF, the core of GFT) synthesizes the next $L$ time steps to form a synthetic window, then performs direct forecasting over the shortened horizon $N - L$ using both observed and generated data.

Let $Y = (X_1, ..., X_M)$ be the observed history. GFT applies generative modeling to predict $(X_{M+1}, ..., X_{M+L})$, denoted $\tilde{Y}_L$, and then the transformer-based predictor forecasts $(X_{M+L+1}, ..., X_{M+N})$. The synthetic window length $L$ serves as a tuning parameter, interpolating between the bias–variance regimes of DF (small $L$) and IF (large $L$).

The theoretical foundation of GenF is a bias–variance decomposition of the mean squared error at horizon $N$:

$\mathrm{MSE}_N = Z(N) + B(N) + V(N)$

where $Z(N)$ is irreducible noise, $B(N)$ is squared bias, and $V(N)$ is variance. For GenF, the joint error is

$$S_{\mathrm{GenF}} = [B_{\mathrm{iter}}(L) + V_{\mathrm{iter}}(L)] + [B_{\mathrm{dir}}(N-L) + V_{\mathrm{dir}}(N-L) + \mathbb{E}_\theta[\gamma(\theta, N-L)^2]]$$

with $\gamma(\theta, N-L)$ quantifying error due to synthetic data usage. Theoretical bounds and recurrence relations (see equations below) rigorously demonstrate that, under standard continuity and statistical assumptions, the GenF framework can strictly reduce upper bounds on forecast error compared to DF or IF alone.

2. Components of the GFT Architecture

(a) Conditional Wasserstein GAN for Time Series (CWGAN-TS)

The CWGAN-TS module generatively synthesizes the next $L$-step window by conditioning on the past $M$ observed inputs. Unlike classical GANs, CWGAN-TS employs a Wasserstein loss with gradient penalty to enforce the 1-Lipschitz constraint, enhancing stability during adversarial training. The generator’s loss includes both unsupervised adversarial (Wasserstein) and supervised $L_2$ penalty terms:

$$\mathcal{L}_S = \mathcal{L}_U + \eta \|X_{M+1} - \bar{X}_{M+1}\|_2$$

where $\mathcal{L}_U$ is the Wasserstein loss (with gradient penalty), and $\eta$ trades off the supervised error.

This conditional architecture ensures that the generated series preserves temporal dynamics, and the supervised stabilization further reduces propagation of generative errors—a major limitation when using LSTM- or GAN-based iterative generation without supervision.

(b) Transformer-Based Predictor

The transformer predictor operates on a sequence comprising both observed data and the synthetic $L$-step window from the CWGAN-TS. It uses standard multi-head self-attention with positional encodings. Unlike deeper transformer variants for longer sequences, the inclusion of synthetic data to “bridge” the forecast gap enables the use of shallower transformer architectures, enabling both high accuracy and reduced parameter counts (15–50% fewer parameters compared to deep transformer baselines).

The self-attention mechanism computes output for each head as:

$$O_h = \mathrm{softmax}\left(\frac{Q_h K_h^T}{\sqrt{d_k}}\right) V_h$$

where $Q_h$, $K_h$, $V_h$ are learned projections of the input window.

(c) Information Theoretic Clustering (ITC) Algorithm

To enhance training on heterogeneous multi-unit datasets, an information theoretic clustering strategy based on mutual information is used. Each unit $P_i$ is scored

$$J(P_i) = \sum_{P_j \in \mathcal{D}, j \neq i} I(P_i ; P_j)$$

with $I(\cdot;\cdot)$ the mutual information estimate. Units are grouped and sampled by these scores, ensuring that CWGAN-TS and the transformer predictor are trained on representative and diverse subsets, improving generalization and reducing sample redundancy. Empirically, this procedure improves synthetic data quality by up to 62% relative to random sampling.

3. Experimental Results

GFT has been empirically validated on five diverse public datasets:

• MIMIC-III Vital Signs
• Multi-Site Air Quality (UCI)
• World Energy Consumption
• Greenhouse Gas Concentrations
• Household Electricity Consumption

Across these datasets, GFT outperformed state-of-the-art methods including TLSTM, LSTNet, DeepAR, Informer, and LogSparse. The approach achieved 5–11% lower mean absolute error and used 15–50% fewer parameters compared to benchmark transformer models. Results further confirm that longer synthetic windows (larger $L$) confer greater improvements as the prediction horizon $N$ increases.

4. Ablation and Component Analyses

Ablation studies demonstrated that:

• The use of the Wasserstein GAN loss (with gradient penalty and supervised error term) in CWGAN-TS is critical; it yielded up to 65% lower synthetic generation error relative to LSTM-based baselines.
• Removing the ITC (random sampling, CWGAN-RS) diminished the quality of synthetic data and subsequent forecast accuracy.
• Direct forecasting ($L=0$) and iterative forecasting ($L=N-1$) represent edge cases of the GFT strategy. BLending the two via GenF (with intermediate $L$) yielded consistently lower forecasting error in practice.

5. Key Mathematical Formulations

Theoretical foundations are expressed via the following formulas:

Equation Type	Mathematical Expression
Bias–Variance Decomposition	$$\mathrm{MSE}_N = \mathbb{E}_Y[(X_{M+N} - u_{M+N})^2|Y] + [u_{M+N} - \mathbb{E}_\theta f(Y, \theta, N)]^2 + \mathbb{E}_{Y,\theta}[ (f(Y, \theta, N) - \mathbb{E}_\theta f(Y, \theta, N))^2 ]$$
GFT Error Decomposition	$$S_{\mathrm{GenF}} = [B_{\mathrm{iter}}(L) + V_{\mathrm{iter}}(L)] + [B_{\mathrm{dir}}(N-L) + V_{\mathrm{dir}}(N-L) + \mathbb{E}_\theta[\gamma(\theta, N-L)^2]]$$
Recurrence for Error Upper Bound	$$b_\alpha(k+1) = b_\alpha(k) \cdot (L_1 + 1 + L_2 b_\alpha(k))$$ with $b_\alpha(1) = \alpha \sigma_I^2$
Self-Attention in Transformer Predictor	$$O_h = \mathrm{softmax}\left(\frac{Q_h K_h^T}{\sqrt{d_k}}\right) V_h$$

These formalize how error propagates in GenF and why a judicious mixture of iterative and direct approaches—enabled by synthetic data generation and flexible prediction windowing—reduces expected error bounds.

6. Practical Considerations and Computational Efficiency

The GFT approach confers important practical advantages:

• The ability to tune the synthetic window length $L$ provides flexible control over the bias–variance trade-off and improves robustness to error accumulation.
• The architectural simplicity of CWGAN-TS and a relatively shallow transformer reduces training and inference costs, confirmed experimentally by reduced parameter counts and comparable or superior accuracy.
• The ITC algorithm ensures that the system is robust to uneven sampling and redundant units in multi-entity datasets, which is critical in real-world multi-site and multi-patient forecasting.

Ablation evidence indicates that both the adversarial loss construction and mutual information-based sampling are essential for optimal performance.

7. Summary and Outlook

The Generative Forecasting Transformer (GFT) formalizes a powerful hybrid paradigm for long-range time series forecasting that jointly leverages strong generative modeling (via conditional Wasserstein GANs), temporal self-attention, and representative data selection through mutual information clustering. Theoretical analysis and multi-domain benchmarking evidence demonstrate that GFT achieves lower forecasting error, better bias–variance trade-off, and more efficient parameter usage than leading direct, iterative, and transformer-based forecasting methods (Liu et al., 2021, Liu et al., 2022).

Future directions suggested by these findings include extending the conditional generative modeling to multivariate settings, exploring further improvements in sample selection strategies, and adapting GFT to non-standard time series domains with complex or irregular observation structures.