Wasserstein barycenter regression for estimating the joint dynamics of renewable and fossil fuel energy indices

In order to characterize non-linear system dynamics and to generate term structures of joint distributions, we propose a flexible and multidimensional approach, which exploits Wasserstein barycentric coordinates for histograms. We apply this methodology to study the relationships between the performance in the European market of the renewable energy sector and that of the fossil fuel energy one. Our methodology allows us to estimate the term structure of conditional joint distributions. This optimal barycentric interpolation can be interpreted as a posterior version of the joint distribution with respect to the prior contained in the past histograms history. Once the underlying dynamics mechanism among the set of variables are obtained as optimal Wasserstein barycentric coordinates, the learned dynamic rules can be used to generate term structures of joint distributions.