The sum of forecasts of disaggregated time series is often required to equal the forecast of the aggregate, giving a set of coherent forecasts. The least squares solution for finding coherent forecasts uses a reconciliation approach known as MinT, proposed by Wickramasuriya, Athanasopoulos, and Hyndman (2019). The MinT approach and its variants do not guarantee that the coherent forecasts are non-negative, even when all of the original forecasts are non-negative in nature. This has become a serious issue in applications that are inherently non-negative such as with sales data or tourism numbers. While overcoming this difficulty, we reconsider the least squares minimization problem with non-negativity constraints to ensure that the coherent forecasts are strictly non-negative. The constrained quadratic programming problem is solved using three algorithms. They are the block principal pivoting (BPV) algorithm, projected conjugate gradient (PCG) algorithm, and scaled gradient projection algorithm. A Monte Carlo simulation is performed to evaluate the computational performances of these algorithms as the number of time series increases. The results demonstrate that the BPV algorithm clearly outperforms the rest, and PCG is the second best. The superior performance of the BPV algorithm can be partially attributed to the alternative representation of the weight matrix in the MinT approach. An empirical investigation is carried out to assess the impact of imposing non-negativity constraints on forecast reconciliation over the unconstrained method. It is observed that slight gains in forecast accuracy have occurred at the most disaggregated level. At the aggregated level, slight losses are also observed. Although the gains or losses are negligible, the procedure plays an important role in decision and policy implementation processes.

中文翻译：

---

最佳非负预测对帐

通常需要分解时间序列的预测总和等于总和的预测，从而给出一组连贯的预测。用于找到相干预测的最小二乘解使用了一种称为MinT的对帐方法，该方法由Wickramasuriya，Athanasopoulos和Hyndman（2019）提出。MinT方法及其变体不能保证连贯的预测是非负的，即使所有原始预测本质上都是非负的。在本质上非负的应用程序（例如销售数据或旅游人数）中，这已成为一个严重的问题。在克服这一困难的同时，我们重新考虑了具有非负约束的最小二乘最小化问题，以确保相干预测严格是非负的。使用三种算法解决了约束二次规划问题。它们是块主数据透视（BPV）算法，投影共轭梯度（PCG）算法和缩放梯度投影算法。随着时间序列数量的增加，执行蒙特卡洛模拟以评估这些算法的计算性能。结果表明，BPV算法明显胜过其他算法，而PCG是第二好的算法。BPV算法的优越性能可以部分归因于MinT方法中权重矩阵的替代表示。进行了一项实证研究，以评估对非约束方法施加非负约束对预测对帐的影响。可以观察到，在最细分类的层次上，预测准确性有所提高。在总体水平上，也观察到了轻微的损失。尽管损益微不足道，但该程序在决策和政策实施过程中起着重要作用。