# CRPS Learning

arXiv: https://arxiv.org/abs/2102.00968 · HTML: https://mlsanity.com/paper/2102.00968/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 83 references that are in the map.
- First posted on arXiv 1 Feb 2021.

## Builds on (most cited first, 12 of 12)

- [Online Learning: A Comprehensive Survey](https://mlsanity.com/paper/1802.02871/) (2018)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Second-order Quantile Methods for Experts and Combinatorial Games](https://mlsanity.com/paper/1502.08009/) (2015)
- [Lipschitz Adaptivity with Multiple Learning Rates in Online Learning](https://mlsanity.com/paper/1902.10797/) (2019)
- [Optimal learning with Bernstein Online Aggregation](https://mlsanity.com/paper/1404.1356/) (2014)
- [Bayesian model averaging: A systematic review and conceptual classification](https://mlsanity.com/paper/1509.08864/) (2015)
- [Online Hierarchical Forecasting for Power Consumption Data](https://mlsanity.com/paper/2003.00585/) (2020)
- [Efficient online algorithms for fast-rate regret bounds under sparsity](https://mlsanity.com/paper/1805.09174/) (2018)
- [Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts](https://mlsanity.com/paper/1912.07048/) (2019)
- [Kalman Recursions Aggregated Online](https://mlsanity.com/paper/2002.12173/) (2020)
- [A Multi-model Combination Approach for Probabilistic Wind Power Forecasting](https://mlsanity.com/paper/1702.03613/) (2017)
- [Sequential Aggregation of Probabilistic Forecasts -- Applicaton to Wind Speed Ensemble Forecasts](https://mlsanity.com/paper/2005.03540/) (2020)

## Built on by (most cited first, 21 of 21)

- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [Forecast combinations: an over 50-year review](https://mlsanity.com/paper/2205.04216/) (2022)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Distributional neural networks for electricity price forecasting](https://mlsanity.com/paper/2207.02832/) (2022)
- [Probabilistic forecasting of German electricity imbalance prices](https://mlsanity.com/paper/2205.11439/) (2022)
- [Adaptive Probabilistic Forecasting of Electricity (Net-)Load](https://mlsanity.com/paper/2301.10090/) (2023)
- [Postprocessing of point predictions for probabilistic forecasting of day-ahead electricity prices: The benefits of using isotonic distributional regression](https://mlsanity.com/paper/2404.02270/) (2024)
- [Deep interpretable ensembles](https://mlsanity.com/paper/2205.12729/) (2022)
- [Multivariate Probabilistic CRPS Learning with an Application to Day-Ahead Electricity Prices](https://mlsanity.com/paper/2303.10019/) (2023)
- [Inductive biases in deep learning models for weather prediction](https://mlsanity.com/paper/2304.04664/) (2023)
- [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)
- [Experimental Comparison of Ensemble Methods and Time-to-Event Analysis Models Through Integrated Brier Score and Concordance Index](https://mlsanity.com/paper/2403.07460/) (2024)
- [Improving probabilistic forecasts of extreme wind speeds by training statistical post-processing models with weighted scoring rules](https://mlsanity.com/paper/2407.15900/) (2024)
- [tsGT: Stochastic Time Series Modeling With Transformer](https://mlsanity.com/paper/2403.05713/) (2024)
- [Filling the Missings: Spatiotemporal Data Imputation by Conditional Diffusion](https://mlsanity.com/paper/2506.07099/) (2025)
- [Electricity Price Forecasting: Bridging Linear Models, Neural Networks and Online Learning](https://mlsanity.com/paper/2601.02856/) (2026)
- [Foundation models for electricity price forecasting and battery arbitrage: Can they replace market-specific forecasting models?](https://mlsanity.com/paper/2609.00089/) (2026)
- [Efficient pooling of predictions via kernel embeddings](https://mlsanity.com/paper/2411.16246/) (2024)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)
- [SPECTRA: State-Space Exogenous Context and Temporal-Frequency Resolution Architecture for Probabilistic Energy Forecasting](https://mlsanity.com/paper/2607.20587/) (2026)
- [TEMPER: Temporal Encoder-Masked Probabilistic Ensemble Regressor for Time-Series Forecasting](https://mlsanity.com/paper/2609.23701/) (2026)

## Abstract (arXiv)

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.