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ROmondi
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STST

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∼N(0,2σ2(1−e− ∥s−s′∥
λ
β
)).
Thisleadsto
EG[|Xs−Xs′|p]= 2σ2(1−e− ∥s−s′∥
λ
β
)
p/2
2p/2Γ p+1
2 √π 1F1
−p/2,1/2;− (µs−µs′)2
4σ2(1−e− ∥s−s′∥
λ
β
)
=2pσp 1−e− ∥s−s′∥
λ
β p/2Γ p+1
2 √π 1F1(−p/2,1/2;0)
=2pσp 1−e− ∥s−s′∥
λ
β p/2Γ p+1
2 √π
Finally,
VSp(F,y)=
s,s′∈D
wij(EG[|Xs−Xs′|p]−|ys−ys′|p)2
=
s,s′∈D
wij 2σ2(1−e− ∥s−s′∥
λ
β
)
p/2
2p/2Γ p+1
2 √π −|ys−ys′|p
2
p-VariationScore
pVS(F,y)=
s∈D∗
wsSETp−var,s
(F,y);
=
s∈D∗
ws(EF[Tp−var,s(X)]−Tp−var,s(y))2,
DenoteZ=Xs+(1,1)−Xs+(1,0)−Xs+(0,1)+Xs.ForX∼F,wehaveZ∼N(µZ,σ2
Z)with
µZ=µs+(1,1)−µs+(1,0)−µs+(0,1)+µs=0
45
and
σ2
Z = σ2
s+(1,1) + σ2
s+(1,0) + σ2
s+(0,1) + σ2
s
−2cov(F(s+(1,1)),F(s +(1,0))) −2cov(F(s +(1,1)),F(s +(0,1)) +2cov(F(s+(1,1)),F(s))
+2cov(F(s+(1,0)),F(s +(0,1))) −2cov(F(s +(1,0)),F(s))
−2cov(F(s+(0,1)),F(s))
=4σ2(1+e−(√2/λ)β −2e−(1/λ)β)
Using (24), this leads to
EF[Tp−var,s(X)] = 4σ2(1 +e−(√2/λ)β −2e−(1/λ)β) p/2
2p/2Γ p+1
√
2
π 1F1(−p/2,1/2;0)
= 4σ2(1+e−(√2/λ)β −2e−(1/λ)β) p/2
2p/2Γ p+1
√
2
π
Finally,
pVS(F,y) =
=
wsSETp−var,s
(F,y)
s∈D∗
s∈D∗
ws 4σ2(1+e−(√2/λ)β −2e−(1/λ)β) p/2
2p/2Γ p+1
CRPS of spatial mean
The CRPS of spatial mean is defined as
CRPSmeanP,wP
(F,y) =
√
2
π −|ys+(1,1) −ys+(1,0) −ys+(0,1) + ys|p
wPCRPSmeanP
(F,y)
P∈P
=
P∈P
wPCRPS(meanP(F),meanP(y)),
where P is an ensemble of spatial patches and wP is the weight associated with a patch P ∈ P. The mean
of Gaussian marginals follows a Gaussian distribution :