Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices
This work addresses the challenge of accurate probabilistic forecasting for electricity markets, offering an incremental advancement in online learning methods for multivariate settings.
The paper tackles the problem of combining multivariate probabilistic forecasts by extending the CRPS learning framework to handle dependencies between quantiles and marginals, resulting in significant improvements over uniform combination in forecasting day-ahead electricity prices, as measured by the continuous ranked probability score (CRPS).
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.