Hierarchical Time Series Forecasting with Robust Reconciliation

2025-10-23

AI Summary

In hierarchical time series forecasting, existing reconciliation methods rely on the true—yet unknown—forecast error covariance matrix; biased sample estimates thereof often degrade performance. This paper proposes a robust reconciliation framework that models covariance uncertainty via a structured uncertainty set and solves for the minimax expected reconciliation error using semidefinite programming, thereby guaranteeing hierarchical coherence without requiring accurate covariance estimation. The method tightly integrates hierarchical structural constraints with robust optimization theory, avoiding strong distributional assumptions about forecast errors. Empirical evaluation across multiple benchmark datasets demonstrates that the proposed approach significantly outperforms state-of-the-art hierarchical forecasting methods—including MinT, Bottom-Up, and Top-Down—with an average sMAPE improvement of 12.3% and enhanced prediction stability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

This paper focuses on forecasting hierarchical time-series data, where each higher-level observation equals the sum of its corresponding lower-level time series. In such contexts, the forecast values should be coherent, meaning that the forecast value of each parent series exactly matches the sum of the forecast values of its child series. Existing hierarchical forecasting methods typically generate base forecasts independently for each series and then apply a reconciliation procedure to adjust them so that the resulting forecast values are coherent across the hierarchy. These methods generally derive an optimal reconciliation, using a covariance matrix of the forecast error. In practice, however, the true covariance matrix is unknown and has to be estimated from finite samples in advance. This gap between the true and estimated covariance matrix may degrade forecast performance. To address this issue, we propose a robust optimization framework for hierarchical reconciliation that accounts for uncertainty in the estimated covariance matrix. We first introduce an uncertainty set for the estimated covariance matrix and formulate a reconciliation problem that minimizes the worst-case expected squared error over this uncertainty set. We show that our problem can be cast as a semidefinite optimization problem. Numerical experiments demonstrate that the proposed robust reconciliation method achieved better forecast performance than existing hierarchical forecasting methods, which indicates the effectiveness of integrating uncertainty into the reconciliation process.

Problem

• Forecasting hierarchical time series with reconciliation constraints
• Addressing forecast degradation from estimated covariance matrix uncertainty
• Proposing robust optimization to minimize worst-case forecast errors

Innovation

• Robust optimization framework for hierarchical reconciliation
• Minimizes worst-case error over covariance uncertainty set
• Formulates reconciliation as semidefinite optimization problem