The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS–RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.

中文翻译：

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预测验证：关联确定性和概率指标

预测验证的原理在确定性和概率验证指标之间有很大不同：一般来说，确定性指标衡量差异，而概率指标评估预测分布的可靠性和锐度。本文考虑了均方根误差 (RMSE)，它可以看作是一个确定性度量，以及概率度量连续排名概率得分 (CRPS)，并证明在某些条件下，CRPS 可以用数学方式表示汇总这些指标时的 RMSE。所需条件之一是分布的正态性。另一个条件是，虽然预测集合不需要校准，但任何偏差或过度/不足都不能依赖于预测分布本身。在这些条件下，CRPS 是 RMSE 的一小部分，该部分仅取决于集合散布的异方差性和校准度量。在理想化二维正压湍流的模拟上测试了在完美集合可靠性情况下导出的 CRPS-RMSE 关系。结果表明，尽管不满足常态条件，但这种关系大致成立。