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More details are available in the main text.
In Figure 2, the ES and the patched ES were computed using samples from the forecasts since closed
expressions could not be derived. However, closed formulas for the VS and the pVS were derived and are
available in Appendix E. As already shown in Scheuerer and Hamill (2015), the VS is able to significantly
discriminate misspecification of the dependence structure induced by the range parameter λ (see Fig. 2a).
Smaller orders of p (such as p = 0.5) appear as more informative than higher ones. Moreover, it is able to dis
criminate misspecification induced by the smoothness parameter β (significantly for all orders p considered)
even if it is less marked than for the misspecification of the range λ.
Figure 2b compares the forecasts using the p-variation score with p ∈ {0.5,1,2}. Note that the forecasts
are provided in the same order as in the other sub-figures. The pVS is able to (significantly) discriminate
all four sub-efficient forecasts from the ideal forecast at all order p. In the cases considered, the pVS has a
stronger discriminating ability than the VS; in particular, for misspecification of the smoothness parameter β.
The overall improvement in the discrimination ability of the pVS compared to the VS is due to the fact that it
only considers local pair interactions between grid points; which in the experimental setup considered greatly
improves the signal-to-noise ratio compared to the VS. For example, it would be incapable of differentiating
two forecasts that only differ in their longer-range dependence structure, where the VS should discriminate
the two forecasts.
Figure 2c shows that the patched ESs have a better discrimination ability than the ES. As expected by
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the clear analogy between the variogram score weights and the selection of valid patches, focusing on smaller
patches improves the signal-to-noise ratio. For all patch size s considered, the patched ES significantly
discriminates the ideal forecast from the others. Whereas the ES does not significantly discriminate the
misspecification of smoothness of the under-smooth and over-smooth forecasts. Nonetheless, the patched
ES remains less sensitive than the VS to misspecifications in the dependence structure through the range
parameter λ or the smoothness parameter β.
The VSrelies on the aggregation and transformation principles and is able to discriminate the dependence
structure. Similarly, the pVS is able to discriminate misspecifications of the dependence structure. Being
based on more local transformations (i.e., p-variation transformation instead of variogram transformation),
it has a greater discrimination ability than the VS in this experimental setup. In addition to this known
application of the aggregation and transformation principles, it has been shown that multivariate transfor
mations can be used to obtain patched scores that, in the case of the ES, lead to an improvement in the
signal-to-noise ratio with respect to the original scoring rule.
5.3 Anisotropy
In this example, we focus on the anisotropy of the dependence structure. We introduce geometric anisotropy
in observations and forecasts via the covariance function in the following way
cov(G(s),G(s′)) = exp − ∥s−s′∥A
λ0
with ∥s −s′∥A = (s−s′)TA(s−s′). The matrix A has the following form :
A= cosθ −sinθ
ρsinθ ρcosθ
with θ ∈ [−π/2,π/2] the direction of the anisotropy and ρ the ratio between the axes.
The observations follow the anisotropic version of the model in Eq. (20) where the covariance function
presents the geometric anisotropy introduced above with λ0 = 3 (as previously) and ρ0 = 2 and θ0 = π/4.
Multiple forecasts are considered that only differ in their prediction of the anisotropy in the model:- the ideal forecast has the same distribution as the observations and is used as a reference;- the small-angle forecast and the large-angle forecast have a correct ratio ρ but an under- and over
estimation of the angle, respectively (i.e., θsmall = 0 and θlarge = π/2);- the isotropic forecast and the over-anisotropic forecast have a ratio ρ = 1 and ρ = 3, respectively, but
a correct angle θ.
Since these forecasts differ only in the anisotropy of their dependence structure, scoring rules not suited
to discriminate the dependence structure would not be able to differentiate them. We compare two proper
scoring rules: the variogram score and the anisotropic scoring rule. The variogram score is studied in two
different settings: one where the weights are proportional to the inverse of the distance and one where
the weights are proportional to the inverse of the anisotropic distance ∥·∥A, which is supposed to be more
informed since it is the quantity present in the covariance function. The anisotropic score (AS) is a scoring
rule based on the framework introduced in Section 3 and, in its general form, it is defined as
AS(F,y) =
whSTiso,h
(F,y) =
h
whS(Tiso,h(F),Tiso,h(y)),
h
(21)
where Tiso,h is a transformation summarizing the anisotropy of a field such as the one introduced in (19).
Additionally, we use a special case of this scoring rule where we do not aggregate across the scales h and
where S is the squared error :
STiso,h
(F, y) = SE(Tiso,h(F),Tiso,h(y)) = ETiso,h(F)[X] − Tiso,h(y) 2 .
(22)
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(a) Variogram score
(b) Anisotropic score for different scales h and aggregated across scales (wh = 1/h)
Figure 3: Expectation of interpretable proper scoring rules focused the dependence structure: (a) the var
iogram score and (b) the anisotropic score, for the ideal forecast (violet), the small-angle forecast (lighter
blue), the large-angle forecast (darker blue), the isotropic forecast (lighter orange) and the over-anisotropic
forecast (darker orange). More details are available in the main text.
We use a transformation similar to the one of (19) where instead the axes are the first and second
bisectors. This leads to the following formula:
Tiso,h(X) = − γX((h,h))−γX((−h,h)) 2
2γX((h,h))2
|D((h,h))| + 2γX((−h,h))2
.
|D((−h,h))|
The choice of this transformation instead of the transformation based on the anisotropy along the abscissa
and ordinate is motivated by the fact that it leads to a clearer differentiation of the forecasts (not shown).
Figure 3a presents the variogram score of order p = 0.5 in its two variants. Both the standard VS and the
informed VS are able to significantly discriminate the ideal forecast from the other forecasts but they have a
weak sensitivity to misspecification of the geometric anisotropy. Even though the informed VS is supposed to
increase the signal-to-noise ratio compared to the standard VS; it is not more sensitive to misspecifications in
the experimental setup considered. Other orders of variograms were tested and worsened the discrimination
ability of both standard and informed VS (not shown).
Figure 3b shows the AS (22) with scales 1 ≤ h ≤ 5 for the different forecasts and the aggregated AS
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(21), where the scales are aggregated with weights wh = 1/h. The anisotropic scores were computed using
samples drawn from the forecasts; this explains why the ideal forecast may appear sub-efficient for some
values of h (e.g., h = 4). As aimed by its construction, the AS is able to significantly distinguish the correct
anisotropy behavior in the dependence structure for values of h up to h = 3 included. For h = 4, the AS
does not significantly discriminate the isotropic forecast and the over-anisotropic forecast from the ideal one.
The fact that h = 1 is the most sensitive to misspecifications is probably caused by the fact that the strength
of the dependence structure decays with the distance (i.e., with h). This shows that the hyperparameter h
plays an important role in having an informative AS (as do the weights and the order p for the variogram
score). For h = 2 in particular, it can be seen that the AS is not sensitive to the misspecification of the
ratio ρ and the angle θ in the same manner. This depends on the degree of misspecification but also on the