Self-Adaptive Quantiles for Precipitation Forecasting

How much rain can we expect in Toulouse on Wednesday next week? It is impossible to provide a precise and definitive answer to this question due to the limited predictability of the atmosphere. Ideally, a forecast would be probabilistic, for example expressed in the form of a probability of, say, having at least some rain. However, for some forecast users and applications, an answer expressed in millimeter of rain per 24 h would be needed. A so-called point-forecast can be the output of a single deterministic model. But with ensemble forecasts at hand, how to summarize optimally the ensemble information into a single outcome? The ensemble mean or quantile forecasts are commonly used and proved useful in certain circumstances. In this study, we suggest a new type of point-forecasts, the crossing-point quantile, and argue that it could be better suited for precipitation forecasting than existing approaches, at least for some users. More precisely, for a well-calibrated predictive distribution, the crossing-point quantile is the optimal forecast in terms of Peirce skill score (and equivalently in terms of area under the relative operating characteristic curve) for any event of interest. Along a theoretical proof, we present an application to daily precipitation forecasting over France and discuss the necessary conditions for optimality.