公平评分函数与最优预测行为
On equitable scoring functions and optimal forecasting behaviour
AI总结：
本文通过最优预报对公平评分函数展开系统分析，引入广义对角评分框架，证明最优预报由预测与气候学累积分布函数的交点表征，揭示公平评分下相同预测分布可能产生不同最优预报的特性。
AI中文摘要：
公平评分函数在气象预报验证领域由来已久，近期因稳定概率空间公平误差（SEEPS）评分被用于评估数值及机器学习气象预测系统的降水预报，重新受到关注。本文通过公平评分函数诱导的最优预报，对其展开系统性分析。我们引入广义对角评分，该评分定义了一类广泛的公平评分函数，在等价意义下包含SEEPS评分、格里里（Gerrity）评分、皮尔士（Peirce）评分、巴恩斯顿（Barnston）评分及对角评分。在该框架内，我们证明最优单值预报与分类预报由预测累积分布函数和气候学累积分布函数的交点表征。此外，我们表明广义对角评分在评估预测分布时是恰当的，但并非严格恰当，且对远离交点的重大预报误设定不敏感。该框架还产生了针对分类结果概率预报既恰当又公平的评分规则。对于单值与分类预报，交点表征显示，即使基础预测分布不变，最优预报也可能随气候学条件变化。因此，相同的预测分布在公平评分下可能导致显著不同的最优预报，这对评估此类评分在特定应用中的适用性具有重要意义。
英文摘要：
Equitable scoring functions have a long history in meteorological forecast verification and have recently gained renewed prominence through the use of the Stable Equitable Error in Probability Space (SEEPS) score for evaluating precipitation forecasts from numerical and machine-learning weather prediction systems. This paper provides a systematic analysis of equitable scoring functions through the optimal forecasts they induce. We introduce the generalized diagonal score, which defines a broad class of equitable scoring functions that includes, up to equivalence, the SEEPS, Gerrity, Peirce, Barnston and diagonal scores. Within this framework, we show that optimal single-valued and categorical forecasts are characterized by crossing points between the predictive and climatological cumulative distribution functions. Moreover, we show that the generalized diagonal score, when evaluating predictive distributions, is proper but not strictly proper, and is insensitive to substantial forecast misspecification away from crossing points. The framework also yields scoring rules that are both proper and equitable for probabilistic forecasts with categorical outcomes. For single-valued and categorical forecasts, the crossing-point characterization shows that optimal forecasts can vary across climatologies even when the underlying predictive distribution is unchanged. Consequently, identical predictive distributions may lead to substantially different optimal forecasts under equitable scores, with important implications for assessing the suitability of such scores for any given application.
