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whence the result follows.
T(x)(F −δy)(dx) = EF[T(X)]−T(y)
Rd
C.3 Proof of examples illustrating Proposition 3
Next, we illustrate the Proposition 3 and provide some computations in two cases: the Gaussian kernel
scoring rule and the continuous rank probability score (CRPS).
Gaussian Kernel Scoring Rule. This is the scoring rule related to the Gaussian kernel
ρ(x1,x2) = exp−(x1 −x2)2/2 , x1,x2 ∈ R.
Using a series expansion of the exponential function, we have
ρ(x1,x2) = e−x2
1/2e−x2
2/2
with Tl the transformation defined, for l ≥ 0, by
Tl(x) = 1
(x1x2)l
l!
l≥0
√
=
l! e−x2/2xl.
Tl(x1)Tl(x2)
l≥0
43
As a consequence, the Gaussian kernel scoring rule writes, for all F ∈ P(R) and y ∈ R,
Sρ(F,y) = 1
2 R× R
ρ(x1,x2)(F −δy)(dx1)(F −δy)(dx2)
= 1
2 R× R l≥0
Tl(x1)Tl(x2) (F −δy)(dx1)(F −δy)(dx2)
= 1
2 l≥0 R
Tl(x)(F −δy)(dx) 2
= 1
2 l≥0
EF[Tl(X)]−Tl(y) 2
.
Continuous Ranked Probability Score. The CRPS is the scoring rule with kernel ρ(x1,x2) = |x1| +
|x2|−|x1 −x2|. This kernel is the covariance of the Brownian motion on R and its RKHS is known to be the
Sobolev space H1 = H1(R), see Berlinet and Thomas-Agnan (2004). We recall the definition of the Sobolev
space
H1 = f ∈C(R,R): f(0) =0, ˙ f ∈ L2(R) ,
where ˙ f denotes the derivative of f assumed to be defined almost everywhere and square-integrable. The
inner product on H1 is defined by
⟨f1, f2⟩H1 =
and one can easily check the fundamental relation
⟨ρ(x1,·), ρ(x2,·)⟩H1 =
˙
f1(x) ˙ f2(x)dx
R
˙
ρ(x1,x) ˙ ρ(x2,x)dx = ρ(x1,x2).
R
Here the derivative ˙ ρ(x1,x) = 1[0,x1](x) is taken with respect to the second variable x. Then, we consider
the Haar system defined as the collection of functions
H0
l(x) = H0(x−l) and H1
l,m(x) = 2m/2H1(2mx−l), l ∈ Z, m ≥ 0,
with H0(x) = 1[0,1)(x) and H1(x) = 1[0,1/2)(x)−1[1/2,1)(x). Since the Haar system is an orthonormal basis
of the space L2(R) and the map f ∈ H1 → ˙ f ∈ L2 is an isomorphism between Hilbert spaces, we obtain an
orthonormal basis of H1(R) by considering the primitives vanishing at 0 of the Haar basis functions. Setting
T0(x) = x1[0,1)(x)+1[1,+∞)(x) and T1(x) = 1/2−|x−1/2| 1[0,1](x) the primitive functions of H0 and H1
respectively, we obtain the system
T0
l (x) = T0(x −l), T1
l,m(x) = 2−m/2T1(2mx−l), l ∈ Z, m ≥ 0.
The series representation of the CRPS is then deduced from Proposition 3 and its proof since the collection
{Tl,m: l ∈ Z,m ≥ 0}, is an orthonormal basis of the RKHS associated with the kernel ρ of the CRPS.
D General form of Corollary 1
Corollary 2. Let T = {Ti}1≤i≤m be a set of transformations from Rd to Rk. Let S = {Si}1≤i≤m be a set
of proper scoring rules such that Si is proper relative to Ti(F), for all 1 ≤ i ≤ m. Let w = {wi}1≤i≤m be
nonnegative weights. Then the scoring rule
m
SST,w(F,y) =
is proper relative to F.
i=1
m
wiSiTi
(F,y) =
i=1
wiSi(Ti(F),Ti(y))
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E Scoringrulesofthesimulationstudy
ThefollowingformulasarededucedforaprobabilisticforecastFtakingtheformof theGaussianrandom
fieldmodelofEquation(20).Theformulasoftheaggregatedunivariatescoringrulescanbeobtainedfrom
theformulasinGneitingandRaftery(2007)andJordanetal.(2019)and,thus,arenotpresentedhere.We
focusontheexpressionofthevariogramscoreandtheCRPSofspatialmean.
VariogramScore
VSp(F,y)=
s,s′∈D
wss′ (EF[|Xs−Xs′|p]−|ys−ys′|p)2
ForX∼N(µ,σ2), theabsolutemoment is(Winkelbauer,2014) :
E[|X|ν]=σν2ν/2Γ ν+1
2 √π 1F1 −ν/2,1/2;−µ2
2σ2
, (24)
where1F1 istheconfluenthypergeometricfunctionofthefirstkind.ForX∼F,
Xs−Xs′∼N(µs−µs′,σs2+σs′
2−2cov(Fs,Fs′)