Datasets:
ROmondi
/
STST

Dataset card Data Studio Files
xet
 Community
2
text
stringlengths
1
298
R
(F(z)−G(z))2dz+
R
G(z)(1−G(z))dz
A.6 Dawid-Sebastianiscore
ForanyF,G∈P2(R), theexpectationoftheDawid-Sebastianiscore(9) is:
EG[DSS(F,Y)]= (µF−µG)2
σF2
+σG2
σF2
+2logσF.
Proof.
EG[DSS(F,Y)]=EG
(Y−µF)2
σF2
+2logσF
=EG (Y−µF)2
σF2
+2logσF
NoticingthatEG (Y−µF)2 =EG[SE(F,Y)],
EG[DSS(F,Y)]= (µF−µG)2+σG2
σF2
+2logσF.
A.7 Error-spreadscore
ForanyF,G∈P4(R), theexpectationoftheerror-spreadscore(10) is:
EG[ESS(F,Y)]= (σG2−σF2)+(µG−µF)2−σFγF(µG−µF) 2
+σG2[2(µG−µF)+(σGγG−σFγF)]2
+σG4(βG−γG2−1),
38
whereµF,σ2
F,γFarethemean,thevarianceandtheskewnessoftheprobabilisticforecastF. Similarly,µG,
σ2
G,γGandβGarethefirstfourcenteredmomentsofthedistributionG.Theproof isavailableinAppendix
BofChristensenetal. (2014).
A.8 Logarithmicscore
ForanyF,G∈P(R)suchthatFandGhaveprobabilitydensityfunctionsintheclassL1(R),theexpectation
ofthelogarithmicscore(11) is:
EG[LogS(F,Y)]=DKL(G||F)+H(F),
whereDKL(G||F) istheKullback-LeiblerdivergencefromFtoGandH(F) istheShannonentropyofF.
Theproof isstraightforwardgiventhattheKullback-LeiblerdivergenceandShannonentropyaredefinedas
DKL(G||F)=
R
g(y)log g(y)
f(y) dy;
H(F)=
R
f(y)log(f(y))dy.
A.9 Hyvärinenscore
ForF,Gsuchthattheirdensitiesf exist,aretwicecontinuouslydifferentiableandsatisfyf′(x)/f(x)→0
as|x|→∞andg′(x)/g(x)→0as|x|→∞, theexpectationoftheHyvärinenscoreis:
EG[HS(F,Y)]=
R
f′(y)2
f(y)2
−2f′(y)g′(y)
f(y)g(y) g(y)dy
=
R
f′(y)
f(y) −g′(y)
g(y)
2
g(y)dy−
R
g′(y)2
g(y)2
g(y)dy
wherethelastformulashowstheentropyoftheHyvärinenscore(secondtermontheright-handside).
Proof.
EG[HS(F,Y)]=E 2f′′(Y)
f(Y) −f′(Y)2
f(y)2
=2
R
f′′(y)
f(y) g(y)dy−
R
f′(y)2
f(y)2
g(y)dy
Integratingbyparttheintegralofthefirsttermontheright-handsideleadsto:
R
f′′(y)
f(y) g(y)dy= f′(y)
f(y) g(y)
+∞
−∞
−
R
f′(y) g′(y)f(y)−g(y)f′(y)
f(y)2
dy
=−
R
f′(y)g′(y)
f(y)g(y) g(y)dy+
R
f′(y)2
f(y)2
g(y)dy