Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices

TL;DR

The paper tackles online aggregation of multivariate probabilistic forecasts by extending CRPS learning to a multivariate setting through horizontal aggregation across quantiles. It introduces two smoothing strategies—dimension-reducing basis matrices and penalized smoothing—to model dependencies among marginals and quantiles within an online Bernstein Online Aggregation framework. The approach delivers fast, adaptive weight updates and demonstrates significant CRPS improvements on 24-dimensional day-ahead electricity price distributions, with a fast C++ implementation in the profoc package. The work also discusses hyperparameter tuning, potential extensions (e.g., Laplacian smoothing, copula integration), and practical implications for operational forecasting in energy markets.

Abstract

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.

Figures (8)

• Figure 1: B-Spline functions for selected placements of the knots concerning the inputs of Algorithm \ref{['algo:knots']}. The center of both figures shows the default case of equidistant knots.
• Figure 2: Most recent weighs of JSU1 calculated using different specifications of Algorithm \ref{['algo:boag_smooth']}
• Figure 3: Correlation plot with Pearson's correlation on the lower triangle and distance correlation on the upper triangle.
• Figure 4: Significance and test statistics of the unconditional coverage test of Kupiec for the full set of experts kupiec1995techniques. This table corresponds to the first row, and the BOA column of Table \ref{['tab:crps']}. The facets present 50% (top) and 90% (bottom) intervals, the symbols indicate significance, and the cells are colored w.r.t. the test statistics. Thereby, the upper limit (dark-red) corresponds to the 0.001 significance level.
• Figure 5: Temporal evolution of the weights of Smooth.Forget Bayesian Online at hour 16 across all 99 probabilities