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Forecast verification: Relating deterministic and probabilistic metrics
Author(s) -
Leung Tsz Yan,
Leutbecher Martin,
Reich Sebastian,
Shepherd Theodore G.
Publication year - 2021
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.4120
Subject(s) - probabilistic logic , mean squared error , metric (unit) , heteroscedasticity , mathematics , normality , statistics , probability distribution , forecast verification , calibration , reliability (semiconductor) , fraction (chemistry) , econometrics , forecast skill , power (physics) , operations management , physics , chemistry , organic chemistry , quantum mechanics , economics
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.