Hierarchical Time Series Forecasting with Bayesian Modeling

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.