Modeling the uncertainty on the covariance matrix for probabilistic forecast reconciliation

2025/06/24 by Chiara Carrara, Carrara, Chiara, Dario Azzimonti +5
Engineering · Decision Sciences · #Fault Detection and Control Systems #Forecasting Techniques and Applications #Reservoir Engineering and Simulation Methods

paper · pdf · doi:10.48550/arxiv.2506.19554

Abstract

In minimum trace (MinT) forecast reconciliation, the covariance matrix of the base forecasts errors plays a crucial role. Typically, this matrix is estimated and then treated as known. This can lead to underestimation of the variance of the predictive distribution. To address the problem, we propose a Bayesian reconciliation model that accounts for the uncertainty in the estimation of the covariance matrix. By adopting an Inverse-Wishart prior and assuming Gaussian residuals, the reconciled predictive distribution follows a multivariate t-distribution, obtained in closed-form, rather than a multivariate Gaussian distribution. We evaluate our method on three tourism-related datasets, including a new publicly available dataset. Empirical results show that our approach consistently improves prediction intervals compared to MinT reconciliation.

Citations

• Conformal Prediction for Hierarchical Data
• Forecast Linear Augmented Projection (FLAP): A free lunch to reduce forecast error variance
• Optimal Forecast Reconciliation with Uncertainty Quantification
• Properties of the reconciled distributions for Gaussian and count forecasts
• Efficient probabilistic reconciliation of forecasts for real-valued and count time series
• Forecast combination based forecast reconciliation: insights and extensions
• Probabilistic forecast reconciliation under the Gaussian framework
• A new tidy data structure to support exploration and modeling of temporal data
• A note on the validity of cross-validation for evaluating autoregressive time series prediction
• Evaluating probabilistic forecasts with scoringRules
• On the Conditional Distribution of the Multivariate t Distribution
• Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC
• The tourism forecasting competition
• Strictly Proper Scoring Rules, Prediction, and Estimation
• Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix

Related

• Probabilistic forecast reconciliation under the Gaussian framework
• Properties of point forecast reconciliation approaches
• Optimal Forecast Reconciliation with Uncertainty Quantification
• Forecast combination based forecast reconciliation: insights and extensions
• Properties of the reconciled distributions for Gaussian and count forecasts