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 will be made available in connection with this paper as an open-source R-Package on CRAN.

翻译：本文提出一种结合（或聚合、集成）多变量概率预测的新方法，通过平滑过程考虑分位数与边际分布之间的依赖性，并支持在线学习。我们讨论了两种平滑方法：基于基矩阵的降维平滑与惩罚平滑。该新型在线学习算法将标准 CRPS 学习框架推广至多变量维度，基于伯恩斯坦在线聚合（BOA）实现，并具备最优渐近学习性质。该方法采用水平聚合策略，即跨分位数聚合。本文深入探讨了算法的可能扩展方向及与现有在线预测组合文献相关的若干嵌套情形。我们将所提出的方法应用于日前电价预测，该预测为24维分布预测。实验表明，所提方法在连续排序概率评分（CRPS）上较均匀组合方法有显著提升。我们讨论了权重与超参数的时间演化规律，并展示了优选模型的简化版本结果。本文配套提供基于C++快速实现的开放源码R语言包，将在CRAN平台发布。