CRPS Learning

TL;DR

This paper introduces a novel adaptive weighting method for probabilistic forecasts that optimizes the continuous ranked probability score (CRPS) by considering performance variations over time and within distributions.

Contribution

It proposes a new pointwise CRPS learning algorithm with theoretical guarantees, using B- and P-Spline estimation techniques for batch and online settings.

Findings

The proposed BOA method achieves optimal convergence in theory.

Simulations confirm the method's effectiveness.

Application to EUA prices demonstrates practical utility.

Abstract

Combination and aggregation techniques can significantly improve forecast accuracy. This also holds for probabilistic forecasting methods where predictive distributions are combined. There are several time-varying and adaptive weighting schemes such as Bayesian model averaging (BMA). However, the quality of different forecasts may vary not only over time but also within the distribution. For example, some distribution forecasts may be more accurate in the center of the distributions, while others are better at predicting the tails. Therefore, we introduce a new weighting method that considers the differences in performance over time and within the distribution. We discuss pointwise combination algorithms based on aggregation across quantiles that optimize with respect to the continuous ranked probability score (CRPS). After analyzing the theoretical properties of pointwise CRPS learning, we discuss B- and P-Spline-based estimation techniques for batch and online learning, based on quantile regression and prediction with expert advice. We prove that the proposed fully adaptive Bernstein online aggregation (BOA) method for pointwise CRPS online learning has optimal convergence properties. They are confirmed in simulations and a probabilistic forecasting study for European emission allowance (EUA) prices.