Modeling the uncertainty on the covariance matrix for probabilistic forecast reconciliation

2025-06-24

AI Summary

This study addresses the miscalibration of prediction intervals in probabilistic forecast reconciliation, arising from uncertainty in estimating the base forecast error covariance matrix. To tackle this, we propose a Bayesian forecast reconciliation framework that explicitly models covariance uncertainty. Specifically, we adopt an Inverse-Wishart prior over the covariance matrix and perform hierarchical Bayesian inference, yielding a closed-form multivariate t-distributed reconciled predictive distribution. This distribution inherently captures both aleatoric and epistemic uncertainty, leading to well-calibrated prediction intervals. Empirical evaluation on hierarchical and grouped time series demonstrates that our method achieves superior probabilistic calibration compared to the classical MinT approach: prediction intervals exhibit more appropriate average width and coverage rates closer to nominal levels, thereby substantially improving the reliability of probabilistic forecasts.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Web data integration and cleaning

Abstract

In forecast reconciliation, the covariance matrix of the base forecasts errors plays a crucial role. Typically, this matrix is estimated, and then treated as known. In contrast, we propose a Bayesian reconciliation model that explicitly accounts for the uncertainty in the covariance matrix. We choose an Inverse-Wishart prior, which leads to a multivariate-t reconciled predictive distribution and allows a completely analytical derivation. Empirical experiments demonstrate that this approach improves the accuracy of the prediction intervals with respect to MinT, leading to more reliable probabilistic forecasts.

Problem

• Model uncertainty in covariance matrix for forecast reconciliation
• Bayesian approach addresses covariance matrix uncertainty
• Improves prediction interval accuracy in probabilistic forecasts

Innovation

• Bayesian model for covariance uncertainty
• Inverse-Wishart prior for analytical derivation
• Multivariate-t predictive distribution improves accuracy