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

The Bier Skill Score (BSS) is given by

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

(1)

where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values and $x_{0}$ is the initial measured values. The BSS ranges between negative infinity and one. A BSS value of 1 indicates a perfect agreement between measured and calculated values. Scores equal to or less than 0 indicates that the mean observed value is as or more accurate than the calculated values.

Nash-Sutcliffe Coefficient

The Nash-Sutcliffe Coefficient (E) is commonly used to assess the predictive power of a model. It is defined as

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

(2)

where where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values and ${\bar {x}}=\langle x_{m}\rangle$. The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.

Root-Mean-Squared Error

The Root-Mean-Squared Error (RMSE) also referred to as Root-Mean-Squared Deviation (RMSD) is defined as

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

(3)

where where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values. The RMSE has the same units as the measured and calculated data.

Normalized-Root-Mean-Squared Error

In order to make comparing different RMSE with different units or scales (lab vs field) several non-dimensional forms of the RMSE have been proposed in literature. Here the Normalized-Root-Mean-Squared Error (NRMSE) is defined as

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

(4)

where ${\text{Range}}(x_{m})=max(x_{m})-min(x_{m})$ is the range of the data. The NRMSE is often expressed in units of percent. Smaller values indicate a better agreement between measured and calculated values.

Mean-Absolute Error

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

(5)

where where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values.

Normalized-Mean-Absolute Error

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

(6)

where where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values.

Correlation coefficient is defined as

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

(7)

where where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values.

Bias

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

(8)

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