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where P denotes a patch and |P| its dimension. Proposition 1 ensures that this transformation can be used
to construct proper scoring rules. The scoring rule involved in the construction has to be univariate, however,
the choice depends on the general properties preferred. For example, the SE would focus on the mean of
the transformed quantity, whereas the AE would target its median. It is worth noting that the total can be
derived by the mean transformation by removing the prefactor
totalP(X) =
Xi.
i∈P
In the case of precipitation, the total is more used than the mean since the total precipitation over a river
basin can be decisive in evaluating flood risk. For example, one could construct an adapted version of the
amplitude component of the SAL method (Wernli et al., 2008; Radanovics et al., 2018) using the SE if
the mean total precipitation is of interest. Gneiting (2011) presents other links between the quantity of
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interest and the scoring rule associated. Similarly, the transformations associated with the minimum and
the maximum over a patch P can be obtained :
minP(X) = min
i∈P
(Xi);
maxP(X) = max
i∈P
(Xi).
The maximum or minimum can be useful when considering extreme events. It can help understand if the
severity of an event is well-captured. For example, as minimum and maximum temperatures affect crop
yields (see, e.g., Agnolucci et al. 2020), it can be of particular interest that a weather forecast within an
agricultural model correctly predicts the minimum and maximum temperatures. After studying the mean,
it is natural to think of the moments of higher order. We can define the transformation associated with the
variance over a patch P as
VarP(X) = 1
|P| i∈P
(Xi −meanP(X))2.
The variance transformation can provide information on the fluctuations over a patch and be used to assess
the quality of the local variability of the forecast. In a more general setup, it can be of interest to use a
transformation related to the moment of order n and the transformation associated follows naturally
Mn,P(X) = 1
|P| i∈P
Xn
i .
More application-oriented transformations are the central or standardized moments (e.g., skewness or kurto
sis). Their transformations can be obtained directly from estimators. As underlined in Heinrich-Mertsching
et al. (2021), since Proposition 1 applies to any transformation, there is no condition on having an unbiased
estimator to obtain proper scoring rules.
Threshold exceedance plays an important role in decision making such as weather alerts. For example,
MeteoSwiss’ heat warning levels are based on the exceedance of daily mean temperature over three consec
utive days (Allen et al., 2023a). They can be defined by the simultaneous exceedance of a certain threshold
and the fraction of threshold exceedance (FTE) is the summary statistic associated.
FTEP,t(X) = 1
|P| i∈P
1{Xi≥t}.
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FTEs can be used as an extension of univariate threshold exceedances and it prevents the double-penalty
effect. FTEs may be used to target compound events (e.g., the simultaneous exceedances of a threshold
at multiple locations of interest). Roberts and Lean (2008) used an FTE-based SE over different sizes of
neighborhoods (patches) to verify at which scale forecasts become skillful. To assess extreme precipita
tion forecasts, Rivoire et al. (2023) introduces scores for extremes with temporal and spatial aggregation
separately. Extreme events are defined as values higher than the seasonal 95% quantile. In the subseasonal
to-seasonal range, the temporal patches are 7-day windows centered on the extreme event and the spatial
patches are square boxes of 150 km × 150 km centered on the extreme event. The final scores are transformed
BS (5) with a threshold of one event predicted across the patch.
Correctly predicting the structure dependence is crucial in multivariate forecasting. Variograms are sum
mary statistics representing the dependence structure. The variogram of order p of the pair (i,j) corresponds
to the following transformation :
γp
ij(X) = |Xi −Xj|p.
As mentioned in the Introduction, using both the transformation and aggregation principles, we can recover
the VS of order p (14) introduced in Scheuerer and Hamill (2015) :
d
VSp(F,y) =
i,j=1
d
wijSEγp
ij
(F,y) =
i,j=1
wij (EF[|Xi − Xj|p] −|yi −yj|)2 .
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Along with the well-known VS of order p, Scheuerer and Hamill (2015) introduced alternatives where the
scoring rule applied on the transformation is the CRPS (6) or the AE (3) instead of the SE (2). As mentioned
previously, under the intrinsic hypothesis of Matheron (1963) (i.e., pairwise differences only depend on the
distance between locations), the weights can be selected to obtain an optimal signal-to-noise ratio. Moreover,
the weights could be selected to investigate a specific scale by giving a non-zero weight to pairs separated
by a given distance.
In the case of spatial forecasts over a grid of size d×d, a spatial version of the variogram transformation
is available :
γi,j(X) = |Xi −Xj|p,
where i,j ∈ D = {1,...,d}2 are the coordinates of grid points. Under the intrinsic hypothesis of Matheron
(1963), the variogram between grid points separated by the vector h can be estimated by :
γX(h) = 1
2|D(h)| i∈D(h)
γi,i+h(X),
where D(h) = {i ∈ D : i+h ∈ D}. This directed variogram can be used to target the verification of the
anisotropy of the dependence structure. The isotropy transformation associated to the distance h can be
defined by
Tiso,h(X) = − γX((h,0))−γX((0,h)) 2
2γX((h,0))2
|D((h,0))| + 2γX((0,h))2
.
|D((0,h))|
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This transformation is the isotropy pre-rank function proposed in Allen et al. (2024). The isotropy trans