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extended to transformations with images in different dimensions and paired with different scoring rules (see
Appendix D).
As we will see in the examples developed in the following section, numerous scoring rules used in the
literature are based on these two principles of aggregation and transformation.
Decomposition of kernel scoring rules. We briefly discuss the link between the transformation and
aggregation principles for scoring rules and the specific class of kernel scoring rules. A kernel on Rd is a
measurable function ρ : Rd × Rd → R satisfying the following two properties:
i) (symmetry) ρ(x1,x2) = ρ(x2,x1) for all x1,x2 ∈ Rd;
ii) (non-negativity)
1≤i≤j≤n aiajρ(xi,xj) ≥ 0 for all x1,...,xn ∈ Rd and a1,...,an ∈ R, for all n ∈ N.
The kernel scoring rule Sρ associated with the kernel ρ is defined on the space of predictive distributions
Pρ = F ∈P(Rd):
by
ρ(x,x)F(dx) < +∞
Sρ(F,y) = EF[ρ(X,y)]− 1
2EF[ρ(X,X′)]− 1
2ρ(y,y),
(15)
where y ∈ Rd and X,X′ are independent random variables following F. Importantly, Sρ is proper on Pρ
and, for an ensemble forecast F = 1
M
M
Sρ(F,y) = 1
M
M
m=1δxm
with M members x1,...,xM, it takes the simple form
ρ(xm,y) − 1
m=1
(16)
2M2
1≤m1,m2≤M
ρ(xm1
,xm2
)− 1
2ρ(y,y),
12
making scoring rules particularly useful for ensemble forecasts.
The CRPS is surely the most widely used kernel scoring rule. Equation (6) shows that it is a associated
with the kernel ρ(x1,x2) = |x1|+|x2|−|x1−x2| (the function |x1−x2| is conditionally semi-definite negative
so that ρ is non-negative). For more details on kernel scoring rules, the reader should refer to Gneiting et al.
(2005) or Steinwart and Ziegel (2021).
The following proposition reveals that a kernel scoring rule can always be expressed as an aggregation of
squared errors (SEs) between transformations of the forecast-observation pair.
Proposition 3. Let Sρ be the kernel scoring rule associated with the kernel ρ. Then there exists a sequence
of transformations Tl : Rd → R, l ≥ 1, such that
Sρ(F,y) = 1
2 l≥1
SE(Tl(F),Tl(y)),
for all predictive distribution F ∈ Pρ and observation y ∈ Rd.
In particular, the series on the right-hand side is always finite. The proof is provided in Appendix C.2 and
relies on the reproducing kernel Hilbert space (RKHS) representation of kernel scoring rules. In particular,
we will see that the sequence (Tl)l≥1 can be chosen as an orthonormal basis of the RKHS associated with
the kernel ρ.
This representation of kernel scoring rules can be useful to understand more deeply the comparison of the
predictive forecast F and observation y. While the definition (15) is quite abstract, the series representation
can be rewritten
Sρ(F,y) =
EF[Tl(X)] −Tl(y) 2
l≥1
with X a random variable following F. In other words, for l ≥ 1, the observed value Tl(y) is compared to
the predicted value Tl(X) under the predictive distribution F using the SE; then all these contributions are
aggregated in a series forming the kernel scoring rule.
To give more intuition, we study two important cases in dimension d = 1. The details of the computations
are provided in Appendix C.3. For the Gaussian kernel scoring rule associated with the kernel
ρ(x1,x2) = exp−(x1 −x2)2/2 ,
some computations yield the series representation
Sρ(F,y) = 1
2 l≥0
1
l! EF[Xle−X2/2] − yle−y2/2 2
so that this score compares the probabilistic forecast F and the observation y through the transforms
Tl(x) = 1
√
l! xle−x2/2, l ≥ 0.
For the CRPS, a possible series representation is obtained thanks to the following wavelet basis of
functions: let T0(x) = x1[0,1)(x) + 1[1,+∞)(x) (plateau function) and T1(x) = 1/2 − |x − 1/2| 1[0,1](x)
(triangle function) and consider the collection of functions
T0
l (x) = T0(x −l), T1
l,m(x) = 2−m/2T1(2mx−l), l ∈ Z,m ≥ 0,
where l ∈ Z is a position parameter and m ≥ 0 a scale parameter. Then, the CRPS can be written as
CRPS(F,y) =
SE(T0
l (F),T0
l (y)) +
l∈Z
=
SE(T1
l,m(F),T1
l,m(y))
l∈Zm≥0
EF[T0(X −l)]−T0(y −l) 2
+
l∈Z
2−mEF[T1(2mX −l)]−T(2my −l) 2
.
l∈Zm≥0
We can see that the CRPS compares forecast and observation through the SE after applying the plateau
and triangle transformations for multiple positions and scales and then aggregates all the contributions.