Tailored postprocessing of ensemble forecasts with distributional regression networks and a quantile-weighted version of the continuous ranked probability score

As a proper score, the continuous ranked probability score (CRPS) is widely used within the field of statistical postprocessing of ensemble forecasts, both for forecast verification and as a loss function for parameter estimation with distributional regression approaches. This includes standard ensemble model output statistics (EMOS) and machine learning (ML) based approaches such as distributional regression networks (DRN). It is known that the CRPS admits equivalent representations as an integral of the Brier score over probability thresholds or an integral of the quantile score over quantile levels. The CRPS can be further generalized with a weighting function to put more weight on certain regions of the predictive distribution (the threshold-weighted CRPS or twCRPS), or to put more weight on certain quantiles of the distribution (quantile-weighting, denoted qwCRPS). In this work, we consider a general 2-parameter class of weight functions that give rise to an analytical expression for the qwCRPS for certain predictive distributions such as the logistic distribution. This generalized version of the CRPS puts a different penalty on over- or underforecasting the meteorological variable, allowing tailored postprocessing for end users with specific cost-loss ratios. We apply a DRN approach using the qwCRPS as loss function to various use cases, including the postprocessing of wind power forecasts for the Belgian Offshore Zone, and compare with the use of the standard CRPS as loss function. We also perform validation using the quantile score and the continuous generalisation of the relative economic value.