Online learning with the Continuous Ranked Probability Score for ensemble forecasting

Ensemble forecasting resorts to multiple individual forecasts to produce a discrete probability distribution that represents the uncertainties accurately. Before every forecast, a weighted empirical distribution function is derived from the ensemble, so as to minimize the Continuous Ranked Probability Score (CRPS). We apply online learning techniques that have previously been used for deterministic forecasting and adapt them for the minimization of the CRPS. The proposed method guarantees theoretically that the aggregated forecast competes, in terms of CRPS, against the best weighted empirical distribution function with weights constant in time. This is illustrated on synthetic data. Additionally, our study improves knowledge of the CRPS expectation for model mixtures. We generalize results about the bias of the CRPS computed with ensemble forecasts and propose a new scheme to achieve fair CRPS minimization, without any assumption about the distribution.