On the calibration of multilevel Monte Carlo ensemble forecasts

The multilevel Monte Carlo method can efficiently compute statistical estimates of discretized random variables for a given error tolerance. Traditionally, only a certain statistic is computed from a particular implementation of multilevel Monte Carlo. This article considers the multilevel case in which one wants to verify and evaluate a single ensemble that forms an empirical approximation to many different statistics, namely an ensemble forecast. We propose a simple algorithm that, in the univariate case, allows one to derive a statistically consistent single ensemble forecast from the hierarchy of ensembles that are formed during an implementation of multilevel Monte Carlo. This ensemble forecast then allows the entire multilevel hierarchy of ensembles to be evaluated using standard ensemble forecast verification techniques. We demonstrate the case of evaluating the calibration of the forecast. Multilevel Monte Carlo (MLMC) can efficiently approximate statistics of forecast distributions for discretized random variables and are formed by a hierarchy of ensembles containing realizations of the random variables discretized on different levels of resolution. One can generate a single ensemble forecast from this hierarchy of ensembles by using a simple algorithm; allowing the evaluation of MLMC approximations using standard ensemble forecast verification techniques, such as the pictured rank histograms that can evaluate the calibration of the forecast.