We encounter time series data in many domains such as finance, physics, business, and weather. One of the main tasks of time series analysis, one that helps to take informed decisions under uncertainty, is forecasting. Time series are often hierarchically structured, e.g., a company sales might be broken down into different regions, and each region into different stores. In some cases the number of series in the hierarchy is too big to fit in a single model to produce forecasts in relevant time, and a decentralized approach is beneficial. One way to do this is to train independent forecasting models for each series and for some summary statistics series implied by the hierarchy (e.g. the sum of all series) and to pass those models to a reconciliation algorithm to improve those forecasts by sharing information between the series. In this work we focus on the reconciliation step, and propose a method to do so from a Bayesian perspective - Bayesian forecast reconciliation. We also define the common case of linear Gaussian reconciliation, where the forecasts are Gaussian and the hierarchy has linear structure, and show that we can compute reconciliation in closed form. We evaluate these methods on synthetic and real data sets, and compare them to other work in this field.

翻译：我们常在金融、物理、商业和天气等多个领域遇到时间序列数据。时间序列分析的主要任务之一是在不确定性下帮助做出明智决策的预测。时间序列通常具有层次结构，例如，一家公司的销售额可能按不同地区划分，而每个地区又细分为不同门店。在某些情况下，层次结构中的序列数量过多，无法通过单一模型在相关时间内生成预测，因此采用分散化方法更为有利。一种实现方式是针对每个序列以及层次结构所隐含的某些汇总统计序列（例如所有序列的总和）训练独立的预测模型，并将这些模型传递给协调算法，通过在序列间共享信息来改进预测结果。本研究聚焦于协调步骤，并提出一种从贝叶斯视角进行协调的方法——即贝叶斯预测协调。我们还定义了线性高斯协调的常见情形，其中预测服从高斯分布且层次结构具有线性形式，并证明可以解析计算协调结果。我们在合成数据集和真实数据集上评估了这些方法，并与该领域的其他研究进行了比较。