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exceedance of a threshold t. For probabilistic forecasts, the BS is defined as |
BSt(F,y) = ((1 −F(t))−1y>t)2 = (F(t)−1y≤t)2, |
(5) |
where 1−F(t) is the predicted probability that the threshold t is exceeded. The BS is proper relative to P(R) |
but not strictly proper. Binary events (e.g., exceedance of thresholds) are relevant in weather forecasting as |
they are used, for example, in operational settings for decision-making. |
All the scoring rules presented above are proper but not strictly proper since they only discriminate |
against specific summary statistics instead of the whole distribution. Nonetheless, they are still used as they |
allow forecasters to verify specific characteristics of the forecast: the mean, the median, the quantile of level |
α or the exceedance of a threshold t. The simplicity of these scoring rules makes them interpretable, thus |
making them essential verification tools. |
Some univariate scoring rules contain a summary statistic: for example, the formulas of the QS (4) or |
the BS (5) contain the exceedance of a threshold t and the quantile of level α, respectively. They can be |
seen as a scoring function applied to a summary statistic. This duality can be understood through the link |
between scoring functions and scoring rules through consistent functionals as presented in Gneiting (2011) |
or Section 2.2 in Lerch et al. (2017). |
Other summary statistics can be of interest depending on applications. Nonetheless, it is worth not |
ing that mispecifications of numerous summary statistics cannot be discriminated because of their non |
elicitability. Non-elicitability of a transformation implies that no proper scoring rule can be constructed |
such that efficient forecasts are forecasts where the transformation is equal to the one of the true distri |
bution. For example, the variance is known to be non-elicitable; however, it is jointly elicitable with the |
mean (see, e.g., Brehmer 2017). Readers interested in details regarding elicitable, non-elicitable and jointly |
elicitable transformations may refer to Gneiting (2011), Brehmer and Strokorb (2019) and references therein. |
A strictly proper scoring rule should discriminate the whole distribution and not only specific summary |
statistics. The continuous ranked probability score (CRPS; Matheson and Winkler 1976) is the most popular |
univariate scoring rule in weather forecasting applications and can be expressed by the following expressions |
CRPS(F,y) = EF|X −y|− 1 |
2EF|X −X′|, |
= |
=2 |
BSz(F,y)dz, |
R |
1 |
0 |
QSα(F,y)dα, |
(6) |
(7) |
(8) |
where y ∈ R and X and X′ are independent random variables following F, with a finite first moment. |
Equations (7) and (8) show that the CRPS is linked with the BS and the QS. Broadly speaking, as the QS |
discriminates a quantile associated with a specific level, integrating the QS across all levels discriminates the |
quantile function that fully characterizes univariate distributions. Similarly, integrating the BS across all |
thresholds discriminates the cumulative distribution function that also fully characterizes univariate distri |
butions. The CRPS is a strictly proper scoring rule relative to P1(R), the class of Borel probability measures |
on R with a finite first moment. In addition, Equation (6) indicates the CRPS values have the same units |
as observations. In the case of deterministic forecasts, the CRPS reduces to the absolute error, in its scor |
ing function form (Hersbach, 2000). The use of the CRPS for ensemble forecast is straightforward using |
expectations as in (6). Ferro et al. (2008) and Zamo and Naveau (2017) studied estimators of the CRPS for |
ensemble forecasts. |
In addition to scoring rules based on scoring functions, some scoring rules use the moments of the |
probabilistic forecast F. The SE (2) depends on the forecast only through its mean µF. The Dawid |
Sebastiani score (DSS; Dawid and Sebastiani 1999) is a scoring rule depending on the forecast F only |
5 |
through its first two central moments. The DSS is expressed as |
DSS(F,y) = 2log(σF)+ (µF −y)2 |
σF2 |
, |
(9) |
where µF and σF2 are the mean and the variance of the distribution F. The DSS is proper relative to P2(R) |
but not strictly proper, since efficient forecasts only need to correctly predict the first two central moments |
(see Appendix A). Dawid and Sebastiani (1999) proposed a more general class of proper scoring rules but |
the DSS, as defined in (9), can be seen as a special case of the logarithmic score (up to an additive constant), |
introduced further down. |
Another scoring rule relying on the central moments of the probabilistic forecast F up to order three is |
the error-spread score (ESS; Christensen et al. 2014). The ESS is defined as |
ESS(F,y) = (σF2 −(µF −y)2 −(µF −y)σFγF)2, |
(10) |
where µF, σ2 |
F and γF are the mean, the variance and the skewness of the probabilistic forecast F. The |
ESS is proper relative to P4(R). As for the other scoring rules only based on moments of the forecast pre |
sented above, the expected ESS compares the probabilistic forecast F with the true distribution only via |
their four first moments (see Appendix A). Scoring rules based on central moments of higher order could be |
built following the process described in Christensen et al. (2014). Such scoring rules would benefit from the |
interpretability induced by their construction and the ease to be applied to ensemble forecasts. However, |
they would also inherit the limitation of being only proper. |
When the probabilistic forecast F has a pdf f, scoring rules of a different type can be defined. Let Lα(R) |
denote the class of probability measures on R that are absolutely continuous with respect to µ (usually taken |
as the Lebesgue measure) and have µ-density f such that |
1/α |
∥f∥α = |
f(x)αµ(dx) |
R |
<∞. |
The most popular scoring rule based on the pdf is the logarithmic score (also known as ignorance score; |
Good 1952; Roulston and Smith 2002). The logarithmic score is defined as |
LogS(F,y) = −log(f(y)), |
(11) |
for y such that f(y) > 0. In its formulation, the logarithmic score is different from the scoring rules seen |
previously. Good (1952) proposed the logarithmic score knowing its link with the theory of information: |
its entropy is the Shannon entropy (Shannon, 1948) and its expectation is related to the Kullback-Leibler |
divergence (Kullback and Leibler, 1951) (see Appendix A). The logarithmic score is strictly proper relative |
to the class L1(R). Moreover, inference via minimization of the expected logarithmic score is equivalent to |
maximum likelihood estimation (see, e.g., Dawid et al. 2015). The logarithmic score belongs to the family of |
local scoring rules, which are scoring rules only depending on y, f(y) and its derivatives up to a finite order. |
Another local scoring rule is the Hyvärinen score (also known as the gradient scoring rule; Hyvärinen 2005) |
and it is defined as |
HS(F,y) = 2f′′(y) |
f(y) − f′(y)2 |
f(y)2 |
, |