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