DeepL\'evy: Learning Heavy-Tailed Uncertainty in Highly Volatile Time Series

TL;DR

DeepLévy introduces a neural framework that models heavy-tailed uncertainty in volatile time series by learning mixtures of Lévý stable distributions through characteristic function matching.

Contribution

It proposes a novel mixture-based approach for Lévý stable distributions that captures complex, non-Gaussian tail behaviors in probabilistic forecasting.

Findings

Outperforms existing models in tail risk metrics.

Effectively models extreme volatility in real and synthetic data.

Adapts to context-dependent heavy-tailed behaviors.

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 L\'evy 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 DeepL\'evy, a neural framework that learns mixtures of L\'evy stable distributions by minimizing the discrepancy between empirical and parametric characteristic functions. DeepL\'evy incorporates a mixture mechanism that adaptively learns context-dependent weights and parameters over multiple L\'evy components, enabling flexible multi-horizon uncertainty modeling. Evaluations on both real and synthetic datasets demonstrate that DeepL\'evy outperforms state-of-the-art deep probabilistic forecasting approaches in tail risk metrics, especially under extreme volatility.