Robust optimal reconciliation for hierarchical time series forecasting with M-estimation

2026-02-26

AI Summary

This study addresses the performance degradation of traditional reconciliation methods in hierarchical time series forecasting when faced with outliers and non-normal errors. To this end, it introduces M-estimation into the hierarchical reconciliation framework for the first time, constructing a robust loss function that reconciles base forecasts while strictly adhering to hierarchical aggregation constraints. The resulting optimization problem is efficiently solved via a local quadratic approximation combined with an improved Newton–Raphson algorithm. The proposed method significantly enhances forecast robustness in the presence of outliers or non-normal errors, while maintaining high efficiency under clean data conditions. Empirical evaluations on both simulated datasets and real-world Australian domestic tourism data demonstrate its consistent superiority over existing approaches, thereby extending the applicability of robust forecasting to complex hierarchical structures.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Machine Learning: Ensemble Methods

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Aggregation constraints, arising from geographical or sectoral division, frequently emerge in a large set of time series. Coherent forecasts of these constrained series are anticipated to conform to their hierarchical structure organized by the aggregation rules. To enhance its resilience against potential irregular series, we explore the robust reconciliation process for hierarchical time series (HTS) forecasting. We incorporate M-estimation to obtain the reconciled forecasts by minimizing a robust loss function of transforming a group of base forecasts subject to the aggregation constraints. The related minimization procedure is developed and implemented through a modified Newton-Raphson algorithm via local quadratic approximation. Extensive numerical experiments are carried out to evaluate the performance of the proposed method, and the results suggest its feasibility in handling numerous abnormal cases (for instance, series with non-normal errors). The proposed robust reconciliation also demonstrates excellent efficiency when no outliers exist in HTS. Finally, we showcase the practical application of the proposed method in a real-data study on Australian domestic tourism.

Problem

• hierarchical time series
• forecast reconciliation
• aggregation constraints
• robustness
• M-estimation

Innovation

• robust reconciliation
• hierarchical time series
• M-estimation
• aggregation constraints
• Newton-Raphson algorithm