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.

翻译：本文提出了一种新的方法，用于组合（或聚合或集成）多元概率预测，通过一种允许在线学习的平滑过程，考虑分位数与边际分布之间的依赖关系。我们讨论了两种平滑方法：基于基矩阵的降维方法和惩罚平滑方法。这一新的在线学习算法将标准CRPS学习框架推广至多元维度。该算法基于伯恩斯坦在线聚合（BOA），具有最优渐近学习性质。该过程使用水平聚合，即跨分位数的聚合。我们深入探讨了该算法的可能扩展形式，以及与现有在线预测组合文献相关的若干嵌套特例。我们将所提方法应用于日前电价预测，该预测是24维分布预测。与均匀组合相比，所提方法在连续排序概率分数（CRPS）方面取得了显著改进。我们讨论了权重和超参数的时间演化，并展示了偏好模型的简化版本结果。该算法的快速C++实现已提供于CRAN上的开源R包profoc中。