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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 initial measured values <math>x_0</math>, final observed or measured values <math>x_m</math> and final calculated values <math>x_c</math>, 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(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}}
{{Equation|<math> 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 } </math>|2=1}}


*Root-Mean-Squared Error  
*Root-Mean-Squared Error  
{{Equation|<math> RMSE(x,y) = \sqrt{ \bigg\langle \big(  x - y \big)^2  \bigg\rangle } </math>|2=2}}
{{Equation|<math> RMSE = \sqrt{ \bigg\langle \big( x_m - x_c  \big)^2  \bigg\rangle  } </math>|2=2}}
 
*Relative-Root-Mean-Squared Error
{{Equation|<math> RRMSE = \frac{\sqrt{ \bigg\langle \big( x_m - x_c  \big)^2 \bigg\rangle}}{x_m} </math>|2=2}}
 
*Normalized-Root-Mean-Squared Error
{{Equation|<math>  RRMSE = \frac{\sqrt{ \bigg\langle \big( x_m - x_c \big)^2  \bigg\rangle}}{range(x_m)} </math>|2=2}}


*Mean-Absolute Error  
*Mean-Absolute Error  
{{Equation|<math> MAE(x,y) =  \bigg\langle \big| x - y \big|  \bigg\rangle  </math>|2=5}}
{{Equation|<math> MAE =  \bigg\langle \big| x_m - x_c \big|  \bigg\rangle  </math>|2=5}}


*Relative-Mean-Absolute Error  
*Relative-Mean-Absolute Error  
{{Equation|<math>  RMAE(x,y) = \frac{MAE(x,y)}{ \big| x \big| }  </math>|2=5}}
{{Equation|<math>  NMAE = \frac{MAE}{ \big| x_m \big| }  </math>|2=5}}
 
*Normalized-Mean-Absolute Error
{{Equation|<math>  NMAE = \frac{MAE}{ \big| range(x_m)) \big| }  </math>|2=5}}


*Correlation coefficient is defined as  
*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}}
{{Equation|<math>  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} }  </math>|2=7}}


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


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

Revision as of 17:41, 1 June 2011

Given the initial measured values , final observed or measured values and final calculated values , 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
  (1)
  • Root-Mean-Squared Error
  (2)
  • Relative-Root-Mean-Squared Error
  (2)
  • Normalized-Root-Mean-Squared Error
  (2)
  • Mean-Absolute Error
  (5)
  • Relative-Mean-Absolute Error
  (5)
  • Normalized-Mean-Absolute Error
  (5)
  • Correlation coefficient is defined as
  (7)

The bias is given by

  (8)

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