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.

翻译：聚合约束常因地理或部门划分而广泛存在于时间序列集合中。这些受约束序列的一致性预测需遵循由其聚合规则组织的层次结构。为增强对潜在不规则序列的鲁棒性，本研究探索层次时间序列预测的鲁棒协调过程。我们引入M估计方法，通过最小化满足聚合约束的基预测变换的鲁棒损失函数来获得协调预测。相关最小化过程通过局部二次近似的改进牛顿-拉夫森算法实现并执行。通过大量数值实验评估所提方法的性能，结果表明该方法能有效处理多种异常情况（例如具有非正态误差的序列）。即使在无异常值的层次时间序列中，所提出的鲁棒协调方法也展现出卓越的效能。最后，我们通过在澳大利亚国内旅游的实际数据研究中展示了该方法的实际应用价值。