This paper presents a new method for combining (or aggregating or ensembling) multivariate probabilistic forecasts, taking into account dependencies between quantiles and covariates through a smoothing procedure that allows for online learning. Two smoothing methods are discussed: 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. 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. The methodology is applied 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 all discussed methods is provided in the R-Package profoc.

翻译：本文提出了一种组合（或聚合、集成）多元概率预报的新方法，该方法通过平滑过程考虑分位数与协变量之间的依赖关系，并支持在线学习。讨论了两种平滑方法：基于基矩阵的降维平滑与惩罚平滑。新在线学习算法将标准CRPS学习框架推广至多元维度，其基于Bernstein在线聚合（BOA）方法，具有最优渐近学习特性。我们深入探讨了算法的可能扩展方向，以及现有在线预报组合文献中的若干嵌套情形。该方法应用于日前电价预测，即24维分布预报。实验表明，所提方法在连续排序概率评分（CRPS）指标上较均匀组合方法有显著改进。我们分析了权重与超参数的时间演化规律，并展示了优选模型的精简版本结果。所有讨论方法的快速C++实现已集成于R包profoc中。