Probabilistic reconciliation of mixed-type hierarchical time series

Hierarchical time series are collections of time series that are formed via aggregation, and thus adhere to some linear constraints. The forecasts for hierarchical time series should be coherent, i.e., they should satisfy the same constraints. In a probabilistic setting, forecasts are in the form of predictive distributions. Probabilistic reconciliation adjusts the predictive distributions, yielding a joint reconciled distribution that assigns positive probability only to coherent forecasts. There are methods for the reconciliation of hierarchies containing only Gaussian or only discrete predictive distributions; instead, the reconciliation of mixed hierarchies, i.e. mixtures of discrete and continuous time series, is still an open problem. We propose two different approaches to address this problem: mixed conditioning and top-down conditioning. We discuss their properties and we present experiments with datasets containing up to thousands of time series.