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We describe some fast algorithms for reconciling large collections of time series forecasts with aggregation constraints. The constraints arise due to the need for forecasts of collections of time series with hierarchical or grouped structures to add up in the same manner as the observed time series. We show that the least squares approach to reconciling hierarchical forecasts can be extended to more general non-hierarchical groups of time series, and that the computations can be handled efficiently by exploiting the structure of the associated design matrix. Our algorithms will reconcile hierarchical forecasts with hierarchies of unlimited size, making forecast reconciliation feasible in business applications involving very large numbers of time series.

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In applying capture-recapture methods for closed populations to epidemiology, e.g., in the estimation of the size of a diabetes population, one comes up against the problem of list errors due to mistyping or misinformation. This problem has been studied for just two lists by Seber, Huakau, and Simmons (2000, Biometrics 56, 1227 1232) using the concept of tag loss borrowed from animal population studies. In this article, we discuss a similar method that can be extended to an arbitrary number of lists. The methods are applied to an example.