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

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

(1)

The Root-Mean-Squared Error is defined as

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

(2)

The Relative-Mean-Absolute Error is defined as

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

(3)

The correlation coefficient is defined as

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

(4)

The bias is given by

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

(5)

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