  This paper presents a new method for combining (or aggregating or ensembling)
multivariate probabilistic forecasts, considering dependencies between
quantiles and marginals through a smoothing procedure that allows for online
learning. We discuss two smoothing methods: dimensionality reduction using
Basis matrices and penalized smoothing. The new online learning algorithm
generalizes the standard CRPS learning framework into multivariate dimensions.
It is based on Bernstein Online Aggregation (BOA) and yields optimal asymptotic
learning properties. The procedure uses horizontal aggregation, i.e.,
aggregation across quantiles. We provide an in-depth discussion on possible
extensions of the algorithm and several nested cases related to the existing
literature on online forecast combination. We apply the proposed methodology to
forecasting day-ahead electricity prices, which are 24-dimensional
distributional forecasts. The proposed method yields significant improvements
over uniform combination in terms of continuous ranked probability score
(CRPS). We discuss the temporal evolution of the weights and hyperparameters
and present the results of reduced versions of the preferred model. A fast C++
implementation of the proposed algorithm is provided in the open-source
R-Package profoc on CRAN.
