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example aims to clarify the notion of interpretability and presents notions that will be reused in the following
examples. The verification of marginals, along with other simple quantities, is usually one of the first steps
of any multivariate forecast verification process.
Observations follow the model of (20) and multiple competing forecasts are considered:- the ideal forecast is the Gaussian distribution generating observations and is used as a reference;- the biased forecast is a Gaussian predictive distribution with the same covariance structure as the
observation but a different mean E[Fbias(s)] = c = 0.255;- the overdispersed forecast and the underdispersed forecast are Gaussian predictive distributions from
the same model as the observations except for an overestimation (σ = 1.4) and an underestimation
(σ = 2/3) of the variance respectively;- the location-scale Student forecast is used where the marginals follow location-scale Student-t distri
butions with parameters µ = 0, df = 5, and τ is such that the standard deviation is 0.745 and the
covariance structure the same as in (20).
In order to compare the predictive performance of forecasts, we use scoring rules constructed by aggre
gating univariate scoring rules. Here, the aggregation is done with uniform weights since there is no prior
knowledge on the locations. The univariate scoring rules considered are the continuous ranked probability
score (CRPS), the Brier score (BS), the quantile score (QS), the squared error (SE) and the Dawid-Sebastiani
score (DSS). Figure 1a compares five different forecasts based on their expected CRPS. It can be seen that
all forecasts except for the ideal one have similar expected values and no sub-efficient forecast is significantly
better than the others. In order to gain more insight into the predictive performance of the forecast, it is
necessary to use other scoring rules. In practice, the distribution is unknown; thus, it is impossible to know
if a forecast is efficient; it is only possible to provide a ranking linked to the closeness of the forecast with
respect to the observations. The definition of closeness depends on the scoring rule used: for example, the
CRPSdefines closeness in terms of the integrated quadratic distance between the two cumulative distribution
functions (see, e.g., Thorarinsdottir and Schuhen 2018).
If the quantity of interest is the value of a quantile of a certain level α, the aggregated QS is an appropriate
scoring rule. Figure 1b shows the expected aggregated QS for three different levels α : α = 0.5, α = .75
and α = 0.95. α = 0.5 is associated with the prediction of the median and, since all the forecasts are
symmetric and only the biased forecast is not centered on zero, the other forecasts are equally the best and
1https://github.com/pic-romain/aggregation-transformation
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(a) Aggregated CRPS
(c) Aggregated BS
(b) Aggregated QS
(d) Aggregated DSS and SE
Figure 1: Expectation of aggregated univariate scoring rules: (a) the CRPS, (b) the quantile score, (c) the
Brier score, and (d) the squared error and the Dawid-Sebastiani score, for the ideal forecast (light violet), a
biased forecast (orange), an under-dispersed forecast (lighter blue), an over-dispersed forecast (darker blue)
and a local-scale Student forecast (green). More details are available in the main text.
efficient forecasts. If the third quartile is of interest (α = 0.75), the location-scale Student forecast appears
as significantly the best (among the non-ideal). For the higher level of α = 0.95, the biased forecast is
significantly the best since its bias error seems to be compensated by its correct prediction of the variance.
Depending on the level of interest, the best forecast varies; the only forecast that would appear to be the
best regardless of the level α is the ideal forecast, as implied by (8).
If a quantity of interest is the exceedance of a threshold t at each location, then the aggregated BS is
an interesting scoring rule. Figure 1c shows the expectation of aggregated BS for the different forecasts and
for two different thresholds (t = 0.5 and t = 1). Among the non-ideal forecasts, there seems to be a clearer
ranking than for the CRPS. The overdispersed forecast is significantly the best regarding the prediction
of the exceedance of the threshold t = 0.5 and the biased forecast is significantly the best regarding the
exceedance of t = 1. As for the aggregated quantile score, the best forecast depends on the threshold t
considered and the only forecast that is the best regardless of the threshold t is the ideal one (see Eq. (7)).
If the moments are of interest, the aggregated SE discriminates the first moment (i.e., the mean) and the
aggregated DSS discriminates the first two moments (i.e., the mean and the variance). Figure 1d presents the
expected values of these scoring rules for the different forecasts considered in this example. The aggregated
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SEs of all forecasts, except the biased forecast, are equal since they have the same (correct) marginal means.
The aggregated DSS presents the biased forecast as significantly the best one (among non-ideal). This is
caused by the combined discrimination of the first two moments of the Dawid-Sebastiani score (see Eq. (9)
and Appendix A).
5.2 Multivariate scores over patches
This second numerical experiment focuses on the prediction of the dependence structure. Observations are
sampled from the model of Eq. (20) and we compare forecasts that differ only in their dependence structure
through misspecification of the range parameter λ and the smoothness parameter β:- the ideal forecast is the Gaussian distribution generating the observations;- the small-range forecast and the large-range forecast are Gaussian predictive distributions from the
same model (20) as the observations except for an underestimation (λ = 1) and an overestimation
(λ = 5), respectively, of the range;- the under-smooth forecast and the over-smooth forecast are Gaussian predictive distributions from the
same model as the observations except for an underestimation (β = 0.5) and an overestimation (β = 2),
respectively, of the smoothness.
Since the forecasts differ only in their dependence structure, scoring rules acting on the 1-dimensional
marginals would not be able to distinguish the ideal forecast from the others. We use the variogram score
(VS) as a reference since it is known to discriminate misspecification of the dependence structure. We
introduce the patched energy score, which results from the aggregation of the ES (with α = 1) over patches,
defined as
ESP,wP
(F,y) =
wPES1(FP,yP),
P∈P
where P is an ensemble of spatial patches, wP is the weight associated with a patch P ∈ P and FP is the
marginal of F over the patch P. In order to make the scoring more interpretable, only square patches of
a given size s are considered and the weights wP are uniform (wP = 1/|P|). Moreover, we consider the
aggregated CRPS and the ES since they are limiting cases of the patched ES for 1×1 patches and a single
patch over the whole domain D, respectively. Additionally, we proposed the p-variation score (pVS), which
is based on the p-variation transformation to focus on the discrimination of the regularity of the random
f
ields,
Tp−var,s(X) = |Xs+(1,1) − Xs+(1,0) − Xs+(0,1) + Xs|p
pVS(F,y) =
=
wsSETp−var,s
(F,y);
s∈D∗
s∈D∗
ws(EF[Tp−var,s(X)] − Tp−var,s(y))2,
where D∗ is the domain D restricted to grid points such that Tp−var,s is defined (i.e., D∗ = {1,...,19} ×
{1,...,19}). Note that in the literature on fractional random fields, the p-variation is an important charac
teristic used to characterize the roughness of a random field and is commonly used for estimation purposes,
see Benassi et al. (2004), Basse-O’Connor et al. (2021) and the references therein.
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(a) Variogram score
(b) p-Variation score
(c) Aggregated CRPS, patched ESs and ES
Figure 2: Expectation of scoring rules focused the dependence structure: (a) the variogram score, (b) the
p-variation score and (c) the patched energy score (and its limiting cases: the aggregated CRPS and the
energy score), for the ideal forecast (violet), the small-range forecast (lighter blue), the large-range forecast
(darker blue), the under-smooth forecast (lighter orange), and the over-smooth forecast (darker orange).