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

$BSS(x,y)=1-\frac{⟨(x-y{)}^2⟩}{⟨(x-x_0{)}^2⟩}$

(1)

• Root-Mean-Squared Error

$RMSE(x,y)=\sqrt{⟨(x-y{)}^2⟩}$

(2)

• Relative-Root-Mean-Squared Error

$RRMSE(x,y,x_0)=\frac{RMSE(x,y)}{RMSE(x,x_0)}$

(3)

• Relative-Root-Mean-Squared Error Score

$RRMSES(x,y,x_0)=1-\frac{RMSE(x,y)}{RMSE(x,x_0)}$

(4)

• Mean-Absolute Error

$MAE(x,y)=⟨|x-y|⟩$

(5)

• Relative-Mean-Absolute Error

$RMAE(x,y)=\frac{MAE(x,y)}{\left|x\right|}$

(5)

• Mean-Absolute Error Score

$MAES(x,y,x_0)=\frac{MAE(x,y)}{MAE(x,x_0)}$

(6)

• Correlation coefficient is defined as

$R=\frac{⟨xy⟩-⟨x⟩⟨y⟩}{\sqrt{⟨x^2⟩-⟨x{⟩}^2}\sqrt{⟨y^2⟩-⟨y{⟩}^2}}$

(7)

The bias is given by

$B=⟨x⟩-⟨y⟩$

(8)

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