This paper presents a formal framework and proposes algorithms to extend forecast reconciliation to discrete-valued data to extend forecast reconciliation to discrete-valued data, including low counts. A novel method is introduced based on recasting the optimisation of scoring rules as an assignment problem, which is solved using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical time series. Two discrete reconciliation algorithms are also proposed and compared against generalisations of the top-down and bottom-up approaches for count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are forecasting criminal offences in Washington D.C. and product unit sales in the M5 dataset. Compared to benchmarks, the proposed framework shows superior performance in both simulations and empirical studies.

翻译：本文提出一个形式化框架，并设计了算法将预测协调方法扩展到离散值数据（包括低计数数据）。我们引入了一种新方法，将评分规则的优化重新表述为分配问题，并通过二次规划求解。该框架可为计数型层次时间序列生成一致的联合概率预测。同时提出两种离散协调算法，并与针对计数数据的自顶向下和自底向上方法的泛化版本进行比较。通过两项模拟实验和两个实证案例验证了所提协调算法能提升预测精度。实证应用包括华盛顿特区犯罪事件预测和M5数据集中的产品销量预测。与基准方法相比，所提框架在模拟和实证研究中均表现出更优性能。