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

$BSS=1-\frac{⟨(x_m-x_c{)}^2⟩}{⟨(x_m-x_0{)}^2⟩}$

(1)

• Root-Mean-Squared Error

$RMSE=\sqrt{⟨(x_m-x_c{)}^2⟩}$

(2)

• Normalized-Root-Mean-Squared Error

$NRMSE=\frac{\sqrt{⟨(x_m-x_c{)}^2⟩}}{\text{Range}(x_m)}$

(3)

• Mean-Absolute Error

$MAE=⟨|x_m-x_c|⟩$

(4)

• Normalized-Mean-Absolute Error

$NMAE=\frac{MAE}{|\text{Range}(x_m)|}$

(5)

• Correlation coefficient is defined as

$R=\frac{⟨x_m{x}_c⟩-⟨x_m⟩⟨x_c⟩}{\sqrt{⟨x_m^2⟩-⟨x_m{⟩}^2}\sqrt{⟨x_c^2⟩-⟨x_c{⟩}^2}}$

(6)

• Bias

$B=⟨x_m⟩-⟨x_c⟩$

(7)

• Nash-Sutcliffe Coefficient

$E=1-\frac{⟨(x_m-x_c{)}^2⟩}{⟨(x_m-\overline{x}{)}^2⟩}$

(8)

where $\overline{x}=⟨x_m⟩$

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