Information‐Driven Modeling of Energy Markets: An Unbalanced Wasserstein Barycenter Approach

Abstract

A novel methodology is proposed for jointly modeling the price dynamics of natural gas and electricity by integrating graph‐based Machine Learning and optimal transport theory. The framework combines visibility graph embeddings with the Wasserstein barycenter to uncover latent structures and asymmetric dependencies between the two interconnected energy markets. Log‐return time series are first transformed into visibility graphs and then embedded into high‐dimensional vector spaces, where complex temporal and structural patterns become more discernible. In the embedding space, an information‐driven Wasserstein barycenter is computed by optimizing the barycenter weights via Shannon entropy maximization. This procedure reveals an asymmetric balance between the two markets, with natural gas exerting a structurally dominant influence. To characterize the joint stochastic dynamics, a Gaussian Mixture Model is fitted to the thus determined unbalanced Wasserstein barycenter using maximum likelihood estimation via the Expectation–Maximization algorithm. An additional Gaussian component is introduced for each commodity to capture market‐specific behavior. The resulting model can be calibrated to match the first four moments of the empirical log‐return distributions and their observed correlation. Applied to Italian market data from 2019 to 2023, a period marked by extreme volatility and systemic shocks, the methodology accurately reproduces both common dynamics and idiosyncratic deviations. The analysis reveals that the entropy‐optimal barycentric weights are for natural gas and for electricity, highlighting a dominant role of the natural gas market in the joint representation. Compared with a comprehensive benchmark of GARCH‐type models, the proposed framework exhibits markedly superior empirical performance. The approach provides a robust, interpretable, and adaptable tool for risk analysis, derivative pricing, and the study of structural interactions across energy markets.