Statistics

Given the observed values x and calculated values y, there are several goodness of fit statistics or skill scores which can be calculated. The definition for some of the more common ones are provided below.

• Brier Skill Score

$BSS(x,y)=1-{\frac {{\bigg \langle }{\big (}x-y{\big )}^{2}{\bigg \rangle }}{{\bigg \langle }{\big (}x-x_{0}{\big )}^{2}{\bigg \rangle }}}$

(1)

• Root-Mean-Squared Error

$RMSE(x,y)={\sqrt {{\bigg \langle }{\big (}x-y{\big )}^{2}{\bigg \rangle }}}$

(2)

• Relative-Root-Mean-Squared Error

$RRMSE(x,y,x_{0})={\frac {RMSE(x,y)}{RMSE(x,x_{0})}}$

(3)

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(3)

• Relative-Root-Mean-Squared Error Score

$RRMSES(x,y,x_{0})=1-{\frac {\sqrt {{\bigg \langle }{\big (}x-y{\big )}^{2}{\bigg \rangle }}}{\sqrt {{\bigg \langle }{\big (}x-x_{0}{\big )}^{2}{\bigg \rangle }}}}$

(4)

• Relative-Mean-Absolute Error

$RMAE(x,y)={\frac {{\bigg \langle }{\big |}x-y{\big |}{\bigg \rangle }}{{\big |}x{\big |}}}$

(5)

• Relative-Mean-Absolute Error Score

$RMAES(x,y,x_{0})=RMAE(x,y)/RMAE(x,x_{0})$

(6)

• Correlation coefficient is defined as

$R={\frac {\langle xy\rangle -\langle x\rangle \langle y\rangle }{{\sqrt {\langle x^{2}\rangle -\langle x\rangle ^{2}}}{\sqrt {\langle y^{2}\rangle -\langle y\rangle ^{2}}}}}$

(7)

The bias is given by

$B=\langle x\rangle -\langle y\rangle$

(8)

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