ROmondi/STST · Datasets at Hugging Face

text stringlengths 1 298
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
,

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

,