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

arXiv: https://arxiv.org/abs/2303.10019 · HTML: https://mlsanity.com/paper/2303.10019/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 65 references that are in the map.
- First posted on arXiv 17 Mar 2023.

## Builds on (most cited first, 11 of 11)

- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Optimal learning with Bernstein Online Aggregation](https://mlsanity.com/paper/1404.1356/) (2014)
- [Distributional neural networks for electricity price forecasting](https://mlsanity.com/paper/2207.02832/) (2022)
- [Day-ahead electricity price forecasting with high-dimensional structures: Univariate vs. multivariate modeling frameworks](https://mlsanity.com/paper/1805.06649/) (2018)
- [CRPS Learning](https://mlsanity.com/paper/2102.00968/) (2021)
- [mlrMBO: A Modular Framework for Model-Based Optimization of Expensive Black-Box Functions](https://mlsanity.com/paper/1703.03373/) (2017)
- [Smoothed Bernstein Online Aggregation for Day-Ahead Electricity Demand Forecasting](https://mlsanity.com/paper/2107.06268/) (2021)
- [Online Aggregation of Probability Forecasts with Confidence](https://mlsanity.com/paper/2109.14309/) (2021)
- [Scores for Multivariate Distributions and Level Sets](https://mlsanity.com/paper/2002.09578/) (2020)
- [Sequential Aggregation of Probabilistic Forecasts -- Applicaton to Wind Speed Ensemble Forecasts](https://mlsanity.com/paper/2005.03540/) (2020)
- [General P-Splines for Non-Uniform B-Splines](https://mlsanity.com/paper/2201.06808/) (2022)

## Built on by (most cited first, 4 of 4)

- [Combining predictive distributions of electricity prices: Does minimizing the CRPS lead to optimal decisions in day-ahead bidding?](https://mlsanity.com/paper/2308.15443/) (2023)
- [Online Multivariate Regularized Distributional Regression for High-dimensional Probabilistic Electricity Price Forecasting](https://mlsanity.com/paper/2504.02518/) (2025)
- [Data-driven Calibration Sample Selection and Forecast Combination in Electricity Price Forecasting: An Application of the ARHNN Method](https://mlsanity.com/paper/2510.15011/) (2025)
- [Downside-Controlled Online Forecast Combination under Delayed and Revised Outcomes](https://mlsanity.com/paper/2609.29096/) (2026)

## Abstract (arXiv)

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.