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

Paper: arXiv 2303.10019

Authors: Jonathan Berrisch, Florian Ziel

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical extension of Bernstein Online Aggregation to multivariate probabilistic forecasts using CRPS learning. It features rigorous empirical application to 24-dimensional electricity price forecasting and excels in reproducibility by providing a fast C++ implementation in an open-source R package.

Research Flowchart

  flowchart TD
    A[Research Goal: Advance Multivariate Probabilistic Forecast Combination] --> B[Data Input: 24-D Day-Ahead Electricity Price Forecasts]
    B --> C[Smoothing Procedure: Basis Matrices & Penalized Smoothing]
    C --> D[Methodology: Horizontal Aggregation via Bernstein Online Aggregation]
    D --> E[Computation: Online CRPS Learning via 'profoc' R Package]
    E --> F[Key Outcome: Superior Performance over Uniform Combination]
    F --> G[Key Outcome: Optimal Asymptotic Properties & Dynamic Weight Evolution]