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David D. Jackson, Sum Mak, David A. Rhoades, Matthew C. Gerstenberger, Naoshi Hirata, Maria Liukis,
Philip J. Maechling, Anne Strader, Matteo Taroni, Stefan Wiemer, Jeremy D. Zechar, and Jiancang
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Maxime Taillardat. Skewed and mixture of gaussian distributions for ensemble postprocessing. Atmosphere,
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34
Maxime Taillardat, Olivier Mestre, Michaël Zamo, and Philippe Naveau. Calibrated ensemble forecasts
using quantile regression forests and ensemble model output statistics. Monthly Weather Review, 144(6):
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Stéphane Vannitsem, John Bjørnar Bremnes, Jonathan Demaeyer, Gavin R. Evans, Jonathan Flowerdew,
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Jonas Bhend, Markus Dabernig, Lesley De Cruz, Leila Hieta, Olivier Mestre, Lionel Moret, Iris Odak
Plenković, Maurice Schmeits, Maxime Taillardat, Joris Van den Bergh, Bert Van Schaeybroeck, Kirien
Whan, and Jussi Ylhaisi. Statistical postprocessing for weather forecasts: Review, challenges, and avenues
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Heini Wernli, Marcus Paulat, Martin Hagen, and Christoph Frei. Sal—a novel quality measure for the
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Michaël Zamo and Philippe Naveau. Estimation of the continuous ranked probability score with limited
information and applications to ensemble weather forecasts. Mathematical Geosciences, 50(2):209–234,
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Florian Ziel and Kevin Berk. Multivariate forecasting evaluation: On sensitive and strictly proper scoring
rules. 2019.
35
A Expected univariate scoring rules
A.1 Squared Error
For any F,G ∈ P2(R), the expectation of the squared error (2) is :
EG[SE(F,Y )] = (µF −µG)2 +σG2,
where µF is the mean of the distribution F and µG and σG2 are the mean and the variance of the distribution
G.
Proof.
EG[SE(F,Y )] = EG[(µF −Y)2]
=µ2
F −2 µFEG[Y]+EG[Y2]
Using the fact that E[X2] = Var(X)+E[X]2,
EG[SE(F,Y )] = µ2
F −2 µFµG +σ2
G +µ2
G
=(µF −µG)2 +σ2
G
A.2 Quantile Score
For any F,G ∈ P1(R), the expectation of the quantile score of level α (4) is :
F−1(α)
EG[QSα(F,Y)] =
−∞
(F−1(α) −y)G(dy)−α
(F−1(α) −y)G(dy);
R
=EG[QSα(G,Y)]+ (G(F−1(α))−α)(F−1(α)−G−1(α))−
Proof. Inspired by the proof of the propriety of the quantile score in Friederichs and Hense (2008).
EG[QSα(F,Y)] =
=
=
(1y≤F−1(α) − α)(F−1(α) −y)G(dy)