Quantifying Uncertainty for Non-Gaussian Ensembles in Complex Systems

Many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another. In short term climate and weather prediction, examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction, or the lack of information in a particular Gaussian ensemble prediction strategy involving the first and second moments compared with the non-Gaussian ensemble itself. The relative entropy is a natural way to quantify this information. Here a recently developed mathematical theory for quantifying this lack of information is converted into a practical algorithmic tool. The theory involves explicit estimators obtained through convex optimization, principal predictability components, a signal/dispersion decomposition, etc. An explicit computationally feasible family of estimators is developed here for estimating the relative entropy over a large dimensional family of variables through a simple hierarchical strategy. Many facets of this computational strategy for estimating uncertainty are applied here for ensemble predictions for two “toy” climate models developed recently: the Galerkin truncation of the Burgers-Hopf equation and the Lorenz ’96 model.