DeepLévy: Learning Heavy-Tailed Uncertainty in Highly Volatile Time Series

2026-05-11

AI Summary

This work addresses the challenge of accurately capturing extreme abrupt events and their associated uncertainty in heavy-tailed time series, which existing deep probabilistic forecasting models struggle to represent. To this end, the authors propose DeepLévy, a novel neural framework that uniquely integrates mixture modeling with Lévy stable distributions and characteristic function matching. By minimizing the discrepancy between empirical and parametric characteristic functions, the method circumvents the intractability of density integration and enables end-to-end learning. Furthermore, a context-adaptive mechanism is introduced to dynamically adjust component weights and parameters, facilitating flexible multi-step uncertainty quantification. Empirical results demonstrate that DeepLévy significantly outperforms current deep probabilistic forecasting approaches on both real-world and synthetic datasets, with particularly notable improvements in tail risk metrics.

Abstract

Modeling uncertainty in heavy-tailed time series remains a critical challenge for deep probabilistic forecasting models, which often struggle to capture abrupt, extreme events. While Lévy stable distributions offer a natural framework for modeling such non-Gaussian behaviors, the intractability of their probability density functions severely limits conventional likelihood-based inference. To address this, we introduce DeepLévy, a neural framework that learns mixtures of Lévy stable distributions by minimizing the discrepancy between empirical and parametric characteristic functions. DeepLévy incorporates a mixture mechanism that adaptively learns context-dependent weights and parameters over multiple Lévy components, enabling flexible multi-horizon uncertainty modeling. Evaluations on both real and synthetic datasets demonstrate that DeepLévy outperforms state-of-the-art deep probabilistic forecasting approaches in tail risk metrics, especially under extreme volatility.

Problem

• heavy-tailed
• time series
• uncertainty modeling
• extreme events
• volatility

Innovation

• Lévy stable distributions
• characteristic function
• heavy-tailed time series
• deep probabilistic forecasting
• tail risk modeling