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hyperparameters of the AS. The aggregated AS allows us to avoid the selection of a scale h while maintaining
the discrimination ability of the lower values of h.
The anisotropic score is an interpretable scoring rule targeting the anisotropy of the dependence structure.
However, it has the limitation of introducing hyperparameters in the form of the scale h and the axes along
which the anisotropy is measured. Aggregation across scales and axes can circumvent the selection of these
hyperparameters; however, a clever choice of weights will be required to maintain the signal-to-noise ratio.
5.4 Double-penalty effect
In this example, we illustrate in a simple setting how scoring rules over patches can be robust to the double
penalty effect (see Section 2.4). The double-penalty effect is introduced in the form of forecasts that deviate
from the ideal forecast by an additive or multiplicative noise term (i.e., nugget effect). The noises are centered
uniforms such that the forecasts are correct on average but incorrect over each grid point.
Observations follow the model of (20) with the parameters σ0 = 1, λ0 = 3 and β0 = 1. As per usual
the ideal forecast, having the same distribution as the observations, is used as a reference. Additive-noised
forecasts are the first type of forecast introduced to test the sensitivity of scoring rules to the form of the
double-penalty effect (presented above). They differ from the ideal forecast through their marginals in the
following way :
Fadd(s) = N(ϵs,σ2
0),
where ϵs ∼ Unif([−r,r]) is a spatial white noise independent at each location s ∈ D. This has an effect on the
mean of the marginals at each grid point. Three different noise range values are tested r ∈ {0.1,0.25,0.5}.
Similarly, multiplicative-noised forecasts that differ from the ideal forecast through their marginals are in
troduced :
Fmul(s) = N(0,σ2(1 +ηs)2),
where ηs ∼ Unif([−r,r]) and s ∈ D. This has an effect on the variance of the marginals at each grid point
and, thus, on the covariance. The same noise range values are tested r ∈ {0.1,0.25,0.5}.
The aggregated CRPS is a naive scoring rule that is sensitive to the double-penalty effect. We propose
the aggregated CRPS of spatial mean which is defined as
CRPSmeanP,wP
(F,y) =
=
wPCRPSmeanP
(F,y);
P∈P
P∈P
wPCRPS(meanP(F),meanP(y)),
where P is an ensemble of spatial patches, wP is the weight associated with a patch P ∈ P and meanP the
spatial mean over the patch P (17). It is a proper scoring rule, and it has an interpretation similar to the
aggregated CRPS, but the forecasts are only evaluated on the performance of their spatial mean. In order
to make the scoring more interpretable, only square patches of a given size s are considered and the weights
wP are uniform. The size of the patches s can be determined by multiple factors such as the physics of
the problem, the constraints of the verification in the case of models on different scales, or hypotheses on
a different behavior below and above the scale of the patch (e.g., independent and identically distributed;
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(a) Aggregated CRPS and CRPS of spatial mean
(b) Aggregated BS and SE of FTE
Figure 4: Expectation of scoring rules tested on their sensitivity to double-penalty effect : (a) the aggregated
CRPS and the aggregated CRPS of spatial mean, and (b) the aggregated Brier score and the aggregated
squared error of fraction of threshold exceedances, for the ideal forecast (violet), the additive-noised forecasts
(shades of blue), and the multiplicative-noised forecasts (shades of orange). For the noised forecasts, darker
colors correspond to larger values of the range r ∈ {0.1, 0.25, 0.5}. More details are available in the main
text.
Taillardat and Mestre 2020). Note that the aggregated CRPS of spatial mean is equal to the aggregated
CRPS when patches of size s = 1 are considered.
If a quantity of interest is the exceedance of a threshold t, the scoring rule associated with that is the
Brier score (5). We compare the aggregated BS with its counterpart over patches: the aggregated SE of the
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FTE. It is defined as
SEFTEP,t,wP
(F,y) =
=
=
wPSEFTEP,t
(F,y);
P∈P
P∈P
P∈P
wPSEFTEP,t(F),FTEP,t(y)
wP EF[FTEP,t(X)]−FTEP,t(y) 2
where P is an ensemble of spatial patches, wP is the weight associated with a patch P ∈ P and FTEP,t the
fraction of threshold exceedance over the patch P and for the threshold t (18). This scoring rule is proper
and focuses on the prediction of the exceedance of a threshold t via the fraction of locations over a patch
P exceeding said threshold. The resemblance with the Brier score is clear and the aggregated SE of FTE
becomes the aggregated BS when patches of size s = 1 are considered.
In Figure 4, the values of the aggregated SE of FTE have been obtained by sampling the forecasts’
distribution. Figure 4a compares the aggregated CRPS and the aggregated CRPS of spatial mean for
different patch size s. For all the scoring rules, we observe an increase in the expected value with the
increase of the range of the noise r. As expected, the aggregated CRPS is very sensitive to noise in the mean
or the variance and, thus, is prone to the double-penalty effect. The aggregated CRPS of spatial mean is
less sensitive to noise on the mean or the variance. Moreover, different patch sizes allow us to select the
spatial scale below which we want to avoid a double penalty. Given that the distribution of the noise is fixed
in this simulation (i.e., uniform), patch size is related to the level of random fluctuations (i.e., the range r)
tolerated by the scoring rule before significant discrimination with respect to the ideal forecast. It is worth
noting that the range r of the noise leads to a stronger increase in the values of these CRPS-related scoring
rules when the noise is on the mean rather than on the variance.
Figure 4b compares the aggregated BS and the aggregated squared error of fraction of threshold ex
ceedances. For simplicity, we fix the threshold t = 1. The aggregated BS is, as expected, sensitive to noise in
the mean or the variance, and an increase in the range of the noise leads to an increase in the expected value
of the score. The aggregated SE of FTE acts as a natural extension of the aggregated BS to patches and
provides scoring rules that are less sensitive to noise on the mean or the variance. The sensitivity evolves
differently with the increase of the patch size s compared to the aggregated CRPS of spatial mean since
the aggregated SE of FTE measures the effect on the average exceedance over a patch. The range r of the
noise apparently leads to a comparable increase in the values of the aggregated SE of FTE when the noise
is additive or multiplicative.
The use of transformations over patches is similar to neighborhood-based methods in the spatial verifica
tion tools framework. Even though avoiding the double-penalty effect is not restricted to tools that do not
penalize forecasts below a certain scale, this simulation setup presents a type of test relevant to probability
forecasts. The patched-based scoring rules proposed here are not by themselves a sufficient verification tool
since they are insensitive to some unrealistic forecast (e.g., if the mean value over the patch is correct but
deviations may be as large as possible and lead to unchanged values of the scoring rule). As for any other
scoring rule, they should be used with other scoring rules.