While forecast reconciliation has seen great success for real valued data, the method has not yet been comprehensively extended to the discrete case. This paper defines and develops a formal discrete forecast reconciliation framework based on optimising scoring rules using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical time series. Two discrete reconciliation algorithms are proposed and compared to generalisations of the top-down and bottom-up approaches to 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 to forecast criminal offences in Washington D.C. and the exceedance of thresholds in age-specific mortality rates in Australia. Compared to the top-down and bottom-up approaches, the proposed framework shows superior performance in both simulations and empirical studies.

翻译：虽然预测协调在实值数据领域取得了显著成功，但该方法尚未全面扩展到离散情形。本文定义并发展了一个基于二次规划优化评分规则的正式离散预测协调框架。该框架为计数层次时间序列生成连贯的联合概率预测。我们提出两种离散协调算法，并将其与自上而下和自下而上方法在计数数据上的推广进行对比。通过两项模拟实验和两项实证案例验证了所提协调算法能提升预测精度。实证应用包括预测华盛顿特区刑事犯罪数量以及澳大利亚年龄特异性死亡率阈值超越事件。相较于自上而下和自下而上的方法，所提框架在模拟研究和实证分析中均展现出更优性能。