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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.
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
*Brier Skill Score
{{Equation|<math> BSS = 1 - \frac{\bigg\langle \big(x-y\big)^2 \bigg\rangle}{\bigg\langle \big(x-x_0\big)^2 \bigg\rangle } </math>|2=1}}
{{Equation|<math> BSS(x,y) = 1 - \frac{\bigg\langle \big(x-y\big)^2 \bigg\rangle}{\bigg\langle \big(x-x_0\big)^2 \bigg\rangle } </math>|2=1}}


The Root-Mean-Squared Error is defined as  
*Root-Mean-Squared Error is defined as  
{{Equation|<math> RMSE = \sqrt{ \bigg\langle \big(  x - y  \big)^2  \bigg\rangle  } </math>|2=2}}
{{Equation|<math> RMSE(x,y) = \sqrt{ \bigg\langle \big(  x - y  \big)^2  \bigg\rangle  } </math>|2=2}}


The Relative-Mean-Absolute Error is defined as
*Relative-Root-Mean-Squared Error  
{{Equation|<math> RMAE = \frac { \bigg\langle \big|  x - y \big|  \bigg\rangle }{ \big| x \big| }  </math>|2=3}}
{{Equation|<math> RRMSE(x,y,x_0) = RMSE(x,y)/RMSE(x,x0) </math>|2=3}}


The correlation coefficient is defined as  
*Relative-Root-Mean-Squared Error Score
{{Equation|<math>  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} }  </math>|2=4}}
{{Equation|<math> RMSES(x,y,x_0 = 1-RRMSE(x,y,x_0)  </math>|2=4}}
 
*Relative-Mean-Absolute Error
{{Equation|<math> RMAE(x,y) =  \frac { \bigg\langle \big| x - y \big|  \bigg\rangle }{ \big| x \big| }  </math>|2=5}}
 
*Relative-Mean-Absolute Error Score
{{Equation|<math> RMAES(x,y,x_0) = RMAE(x,y)/RMAE(x,x_0) </math>|2=6}}
 
*Correlation coefficient is defined as  
{{Equation|<math>  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} }  </math>|2=7}}


The bias is given by
The bias is given by
{{Equation|<math>  B =  \langle x \rangle - \langle y  \rangle  </math>|2=5}}
{{Equation|<math>  B =  \langle x \rangle - \langle y  \rangle  </math>|2=8}}


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[[CMS#Documentation_Portal | Documentation Portal]]
[[CMS#Documentation_Portal | Documentation Portal]]

Revision as of 18:07, 6 December 2010

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−⟨(x−y)2⟩⟨(x−x0)2⟩ (1)
  • Root-Mean-Squared Error is defined as
  RMSE(x,y)=⟨(x−y)2⟩ (2)
  • Relative-Root-Mean-Squared Error
  RRMSE(x,y,x0)=RMSE(x,y)/RMSE(x,x0) (3)
  • Relative-Root-Mean-Squared Error Score
  RMSES(x,y,x0=1−RRMSE(x,y,x0) (4)
  • Relative-Mean-Absolute Error
  RMAE(x,y)=⟨|x−y|⟩|x| (5)
  • Relative-Mean-Absolute Error Score
  RMAES(x,y,x0)=RMAE(x,y)/RMAE(x,x0) (6)
  • Correlation coefficient is defined as
  R=⟨xy⟩−⟨x⟩⟨y⟩⟨x2⟩−⟨x⟩2⟨y2⟩−⟨y⟩2 (7)

The bias is given by

  B=⟨x⟩−⟨y⟩ (8)

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