MVG-CRPS: A Robust Loss Function for Multivariate Probabilistic Forecasting

TL;DR

This paper introduces MVG-CRPS, a new loss function for multivariate Gaussian probabilistic forecasting that enhances robustness and accuracy by reducing sensitivity to outliers, and integrates easily with neural network models.

Contribution

The paper proposes MVG-CRPS, a proper scoring rule with a closed-form expression for multivariate Gaussian distributions, improving robustness and performance in neural probabilistic forecasting.

Findings

MVG-CRPS improves forecasting robustness against outliers.

The method enhances accuracy in multivariate and univariate forecasting tasks.

It provides better uncertainty quantification in probabilistic models.

Abstract

Multivariate Gaussian (MVG) distributions are central to modeling correlated continuous variables in probabilistic forecasting. Neural forecasting models typically parameterize the mean vector and covariance matrix of the distribution using neural networks, optimizing with the log-score (negative log-likelihood) as the loss function. However, the sensitivity of the log-score to outliers can lead to significant errors in the presence of anomalies. Drawing on the continuous ranked probability score (CRPS) for univariate distributions, we propose MVG-CRPS, a strictly proper scoring rule for MVG distributions. MVG-CRPS admits a closed-form expression in terms of neural network outputs, thereby integrating seamlessly into deep learning frameworks. Experiments on real-world datasets across multivariate autoregressive and univariate sequence-to-sequence (Seq2Seq) forecasting tasks show that MVG-CRPS improves robustness, accuracy, and uncertainty quantification in probabilistic forecasting.