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4 Applications of the transformation and aggregation principles
4.1 Projections
Certainly, the most direct type of transformation is projections of forecasts and observations on their k
dimensional marginals. We denote Ti the projection on the i-th component such that Ti(X) = Xi, for all
X∈Rd. This allows the forecaster to assess the predictive performance of a forecast for a specific univariate
marginal independently of the other variables. If S is an univariate scoring rule proper relative to P(R),
then Proposition 1 leads to STi
being proper relative to P(Rd). This "new" scoring rule can be useful if
a given marginal is of particular interest (e.g., location of high interest in a spatial forecast). However, it
can be more interesting to aggregate such scoring rules across all 1-dimensional marginals. This leads to the
following scoring rule
d
SST,w(F,y) =
i=1
wiSTi
(F,y),
where ST is {STi
}1≤i≤d. This setting is popular for assessing the performance of multivariate forecasts
and we briefly present examples from the literature falling under this setting. Aggregation of CRPS (6)
across locations and/or lead times is common practice for plots or comparison tables with uniform weights
(Gneiting et al., 2005; Taillardat et al., 2016; Rasp and Lerch, 2018; Schulz and Lerch, 2022; Lerch and
Polsterer, 2022; Hu et al., 2023) or with more complex schemes such as weights proportional to the cosine
of the latitude (Ben Bouallègue et al., 2024b). The SE (2) and AE (3) can be aggregated to obtain RMSE
and MAE, respectively (Delle Monache et al., 2013; Gneiting et al., 2005; Lerch and Polsterer, 2022; Pathak
et al., 2022). Bremnes (2019) aggregated QSs (4) across stations and different quantile levels of interest with
uniform weights. Note that the multivariate SE (12) can be rewritten as the sum of univariate SE across
1-marginals: SE(F,y) = ∥µF −y∥2
2 = d
i=1SETi
(F,y).
The second simplest choice is the 2-dimensional case, allowing to focus on pair dependency. We denote
T(i,j) the projection on the i-th and j-th components (i.e., the (i,j) pair of components) such that T(i,j)(X) =
Xi,j = (Xi,Xj). In this setting, S has to be a bivariate proper scoring rule to construct a proper scoring
rule ST(i,j)
. The aggregation of such scoring rules becomes
d
SST,w(F,y) =
i,j=1
i=j
wi,jST(i,j)
(F, y).
As suggested in Scheuerer and Hamill (2015) for the VS (14), the weights wi,j can be chosen appropriately to
optimize the signal-to-noise ratio. For example, in a spatial setting where the dependence between locations
is believed to decrease with the distance separating them, the weights wi,j can be chosen to be proportional
to the inverse of the distance. This bivariate setting is less used in the literature, we present two articles
using or mentioning scoring rules within this scope. In a general multivariate setting, Ziel and Berk (2019)
suggests the use of a marginal-copula scoring rule where the copula score is the bivariate copula energy score
(i.e., the aggregation of the energy scores across all the regularized pairs). To focus on the verification of the
temporal dependence of spatio-temporal forecasts, Ben Bouallègue et al. (2024b) uses the bivariate energy
score over consecutive lead times.
In a more general setup, we consider projection on k-dimensional marginals. In order to reduce the
number of transformation-based scores to aggregate, it is standard to focus on localized marginals (e.g.,
belonging to patches of a given spatial size). Denote P = {Pi}1≤i≤m a set of valid patches (for some
criterion or of a given size) and SP the set of transformation-based scores associated with the projections on
the patches P. Given a multivariate scoring rule S proper relative to P(Rk), we can construct the following
aggregated score :
SSP,w(F,y) =
wPSP(F,y).
P∈P
This construction can be used to create a scoring rule only considering the dependence of localized com
ponents, given that the patches are defined in that sense. The use of patches has similar benefits as the
14
weighting of pairs given a belief on their correlations: obtain a better signal-to-noise ratio and improve the
discrimination of the resulting scoring rule. For example, Pacchiardi et al. (2024) introduced patched energy
scores as scoring rules to minimize in order to train a generative neural network. The patched energy scores
are defined for S = ES and square patches spaced by a given stride. Even though spatial patches may be
more intuitive, it is possible to use temporal or spatio-temporal patches. Patch-based scoring rules appear
as a natural member of the neighborhood-based methods of the spatial verification classification mentioned
in Section 2.4. Given that the patches are correctly chosen (e.g., of a size appropriate to the problem at
hand), patch-based scoring rules are not subject to the double-penalty effect.
As noticeable by the low number of examples available in the literature, aggregation (and plain use)
of scoring rules based on projection in dimension k ≥ 2 is not standard practice, probably because such
projections may lack interpretability. Instead, to assess the multivariate aspects of a forecast, scoring rules
relying on summary statistics are often favored.
4.2 Summary statistics
Summary statistics are a central tool of statisticians’ toolboxes as they provide interpretable and understand
able quantities that can be linked to the behavior of the phenomenon studied. Moreover, their interpretability
can be enhanced by the forecaster’s experience and this can be leveraged when constructing scoring rules
based on them. Summary statistics are commonly present during the verification procedure and this can be
extended by the use of new scoring rules derived from any summary statistic of interest. For example, nu
merous summary statistics can come in handy when studying precipitations over a region covered by gridded
observation and forecasts. Firstly, it is common practice to focus on binary events such as the exceedance of
a threshold (e.g., the presence or absence of precipitation). This can be studied by using the BS (5) on all
1-dimensional marginals as mentioned in the previous subsection but also in a multivariate manner through
the fraction of threshold exceedances (FTE) over patches as presented further. Regarding precipitations,
it is standard to be interested in the prediction of total precipitation over a region or a time period. This
transformation of the field can be leveraged to construct a scoring rule. Finally, it is important to verify
that the spatial structure of the forecast matches the spatial structure of observations. The spatial structure
can be (partially) summarized by the variogram or by wavelet transformations. The predictive performance
for the spatial structure can be assessed by their associated scoring rules: the VS of order p (14) and the
wavelet-based score (Buschow et al., 2019). Other summary statistics can be of interest to the phenomenon
studied, Heinrich-Mertsching et al. (2021) present summary statistics specific to point processes focusing on
clustering and intensity.
The most well-known summary statistic is certainly the mean. In spatial statistics, it can be used to
avoid double penalization when we are less interested in the exact location of the forecast but rather in a
regional prediction. The transformation associated with the mean is
meanP(X) = 1
|P| i∈P
Xi,