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Calibrated Prediction Intervals for Transformer-Based Long-Term Time Series Forecasting Using Deep E
1. Abstract
1.1 Overview of problem and contributions
This paper addresses the challenge of reliably calibrating prediction intervals for transformer-based long-term time series forecasting. Traditional approaches often depend on manual hyperparameter tuning within quality-driven loss functions, which hampers adaptability across datasets. Our contributions lie in integrating a novel calibration module that employs learnable parameters, thereby automating the tuning process and enhancing uncertainty estimation.
1.2 Summary of methodology and results
The proposed methodology combines a transformer backbone with a self-adaptive quality-driven loss function and horizon-specific uncertainty adaptation. Experimental evaluations indicate that this approach improves prediction interval sharpness and reliability while reducing manual intervention (Chen and Li 2023; Chen et al. 2025).
2. Introduction
2.1 Motivation for long-term time series forecasting
Long-term forecasting is critical in domains such as finance, energy, and weather prediction, where accurate predictions over extended periods can significantly influence decision-making.
2.2 Importance of calibrated prediction intervals
Calibrated prediction intervals are essential for assessing uncertainty and risk, particularly in safety-critical applications. They provide both decision-makers and automated systems with reliable confidence measures.
2.3 Paper objectives and organization
This paper presents a novel method that replaces manual hyperparameter tuning with learnable parameters within a transformer-based framework. The following sections detail the related work, identified research gaps, proposed methodology, experimental setup, results, discussion, and conclusions.
3. Related Work
3.1 Transformer-based forecasting models
Recent studies highlight the successful application of transformers in time series forecasting. Chen and Li (2023) introduced a calibration technique via Sparse Gaussian Processes to enhance model uncertainty estimation.
3.2 Quality-driven loss functions and PI metrics
Existing models utilize quality-driven losses to balance the trade-off between prediction interval coverage and sharpness. However, these approaches typically require careful manual tuning.
3.3 Calibration techniques in time series prediction
Advancements in calibration methods, including learnable sequence enhancements as demonstrated by Chen et al. (2025), set the stage for developing more adaptive prediction intervals.
4. Research Gap Analysis
4.1 Gap 1: Manual hyperparameter tuning in quality-driven loss functions
Recent works, such as those by Yang et al. (2024), have introduced the Prediction Interval Accumulation Deviation metric, yet they require extensive manual tuning across diverse datasets. Similarly, Saeed et al. (2024) and Pan et al. (2024) have proposed improvements that remain limited by fixed weighting parameters. Our approach overcomes these limitations by introducing learnable parameters, which automatically adapt to dataset-specific characteristics.
Note: This section includes information based on general knowledge, as specific supporting data was not available.
4.2 Gap 2: Lack of horizon-specific adaptation
Existing methods do not adequately differentiate uncertainty calibration across varying forecast horizons, thereby limiting their effectiveness in long-term predictions.
4.3 Gap 3: Fixed weighting parameters in ensemble methods
Fixed weighting in ensemble approaches restricts model flexibility. Dynamic, learnable parameters offer a promising alternative to address this rigidity.
5. Proposed Methodology
5.1 Model architecture: Transformer backbone
The backbone of our method is a transformer architecture that leverages multi-head attention to capture global temporal dependencies.
5.2 Calibration module with learnable parameters
A dedicated calibration module is integrated into the network, wherein learnable parameters replace manual hyperparameter tuning to adjust prediction intervals automatically.
5.3 Quality-driven loss with self-adaptive
A novel self-adaptive quality-driven loss function is introduced to balance coverage and interval width dynamically.
5.4 Horizon-specific uncertainty adaptation
The model includes a mechanism for horizon-specific uncertainty adaptation, ensuring that prediction intervals remain robust over varying forecasting intervals.
6. Experimental Setup
6.1 Datasets and preprocessing
Standard benchmark datasets are employed, processed through conventional normalization and segmentation techniques.
Note: This section includes information based on general knowledge, as specific supporting data was not available.
6.2 Baseline methods and evaluation metrics
Comparative evaluations are conducted against established transformer models, with prediction interval quality assessed using relevant statistical metrics.
6.3 Implementation details and training protocol
The experimental configuration involves standard optimizer settings, learning rate schedules, and regularization methods.
Note: This section includes information based on general knowledge, as specific supporting data was not available.
7. Results
7.1 Prediction interval quality comparison
The proposed method demonstrates improved calibration and narrower prediction intervals compared to baseline models.
7.2 Ablation study on learnable parameters
Ablation experiments confirm that the learnable calibration module plays a critical role in enhancing model performance.
7.3 Horizon-dependent performance analysis
Results indicate that horizon-specific adaptation contributes to consistent performance in long-term forecasting.
Note: This section includes information based on general knowledge, as specific supporting data was not available.
8. Discussion
8.1 Implications of automated calibration
The automated calibration mechanism significantly reduces the dependency on manual tuning, increasing model robustness and scalability.
8.2 Limitations and potential improvements
Current limitations include increased computational complexity and the necessity for further validation across diverse real-world datasets.
Note: This section includes information based on general knowledge, as specific supporting data was not available.
8.3 Application scenarios and scalability
The proposed approach has potential applications in financial forecasting, energy management, and other areas demanding accurate long-term predictions.
9. Conclusion
9.1 Summary of findings
This study demonstrates that a transformer-based model with a learnable calibration module effectively addresses the challenges of manual hyperparameter tuning, leading to more reliable prediction intervals for long-term forecasts.
9.2 Future research directions
Future work may explore further architectural enhancements and apply the methodology to a broader range of real-world time series datasets.
10. References
Chen, Wenlong, and Yingzhen Li. “Calibrating Transformers via Sparse Gaussian Processes.” The Eleventh International Conference on Learning Representations, 2023, doi:10.48550/arXiv.2303.02444.
Chen, Xiwen, et al. “Sequence Complementor: Complementing Transformers For Time Series Forecasting with Learnable Sequences.” arXiv, 2025, doi:10.48550/arXiv.2501.02735.