Statistics

Given the initial measured values $x_{0}$, final observed or measured values $x_{m}$ and final calculated values $x_{c}$, 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

$BSS=1-{\frac {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}{{\bigg \langle }{\big (}x_{m}-x_{0}{\big )}^{2}{\bigg \rangle }}}$

(1)

• Root-Mean-Squared Error

$RMSE={\sqrt {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}}$

(2)

• Normalized-Root-Mean-Squared Error

$NRMSE={\frac {\sqrt {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}}{{\text{Range}}(x_{m})}}$

(3)

• Mean-Absolute Error

$MAE={\bigg \langle }{\big |}x_{m}-x_{c}{\big |}{\bigg \rangle }$

(4)

• Normalized-Mean-Absolute Error

$NMAE={\frac {MAE}{{\big |}{\text{Range}}(x_{m}){\big |}}}$

(5)

• Correlation coefficient is defined as

$R={\frac {\langle x_{m}x_{c}\rangle -\langle x_{m}\rangle \langle x_{c}\rangle }{{\sqrt {\langle x_{m}^{2}\rangle -\langle x_{m}\rangle ^{2}}}{\sqrt {\langle x_{c}^{2}\rangle -\langle x_{c}\rangle ^{2}}}}}$

(6)

• Bias

$B=\langle x_{m}\rangle -\langle x_{c}\rangle$

(7)

• Nash-Sutcliffe Coefficient

$E=1-{\frac {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}{{\bigg \langle }{\big (}x_{m}-{\bar {x}}{\big )}^{2}{\bigg \rangle }}}$

(8)

where ${\bar {x}}=\langle x_{m}\rangle$

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