Probabilistic forecast reconciliation: theory, algorithm, and applications

Hierarchical time series are common in several applied fields. Forecasts are required to be coherent, that is, to satisfy the constraints given by the hierarchy. The most popular technique to enforce coherence is called reconciliation, which adjusts the base forecasts computed for each time series. However, recent works on probabilistic reconciliation present several limitations. In this thesis, we propose a new approach based on conditioning to reconcile any type of forecast distribution. We then introduce a new algorithm, called Bottom-Up Importance Sampling, to efficiently sample from the reconciled distribution. It can be used for any base forecast distribution: discrete, continuous, or in the form of samples, providing a major speedup compared to the current methods. Experiments on several temporal hierarchies show a clear improvement over base probabilistic forecasts. We then study the effects of reconciliation on the forecast distribution, both from a theoretical viewpoint and using some examples with Bernoulli and Poisson distributions. We also present an application to count time series of extreme events on the Credit Default Swap (CDS) market. Finally, we introduce and study the p-Fourier Discrepancy Functions, a new family of metrics for comparing discrete probability measures.