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15 Appendix E: Goodness of Fit Statistics
=15 Appendix E: Goodness of Fit Statistics=
Brier Skill Score
 
==Brier Skill Score==
 
The Brier Skill Score (BSS) is defined as
The Brier Skill Score (BSS) is defined as
(A1)
 
{{Equation|<math>BSS = 1 - \frac{\langle(x_m - x_c)^2 \rangle}{\langle (x_m - x_0)^2 \rangle} </math> |A1}}
 
where the angled brackets indicate averaging, subscripts m, c, and 0 indi-cate measured, calculated, and initial values, respectively. 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. The following quantifications are used for de-scribing the BSS values: 0.8<BSS<1.0 = excellent, 0.6<BSS<0.8 = good, 0.3<BSS<0.6 = reasonable, 0<BSS<0.3 = poor, BSS<0 = bad.
where the angled brackets indicate averaging, subscripts m, c, and 0 indi-cate measured, calculated, and initial values, respectively. 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. The following quantifications are used for de-scribing the BSS values: 0.8<BSS<1.0 = excellent, 0.6<BSS<0.8 = good, 0.3<BSS<0.6 = reasonable, 0<BSS<0.3 = poor, BSS<0 = bad.
Room Mean Squared Error
 
=Room Mean Squared Error=
 
The Root Mean Squared Error (RMSE) is defined as  
The Root Mean Squared Error (RMSE) is defined as  
(A2)
 
{{Equation|<math>RMSE = \sqrt{\langle (x_c - x_m)^2 \rangle}</math>|A2}}
 
The RMSE has the same units as the measured data. Lower values of RMSE indicate a better match between measured and computed values.  
The RMSE has the same units as the measured data. Lower values of RMSE indicate a better match between measured and computed values.  
Normalized Root Mean Squared Error
 
=Normalized Root Mean Squared Error=
 
The Normalized Root Mean Squared Error (NRMSE) is
The Normalized Root Mean Squared Error (NRMSE) is
(A3)
 
The NRMSE is often expressed in units of percent. The measured data range   can be estimated as . Lower values of NRMSE indicate a better agreement between measured and computed values.  
{{Equation|<math>NRMSE = \frac{RMSE}{range(x_m)}</math>|A3}}
Mean Absolute Error
 
The NRMSE is often expressed in units of percent. The measured data range ''range(x<sub>m</sub>) ''  can be estimated as max(x<sub>m</sub>) - min(x<sub>m</sub>) . Lower values of NRMSE indicate a better agreement between measured and computed values.  
 
=Mean Absolute Error=
 
The Mean Absolute Error (MAE) is defined as
The Mean Absolute Error (MAE) is defined as
(A4)
 
Normalized Mean Absolute Error
{{Equation|<math>MAE = \langle | x_c - x_m | \rangle</math>|A4}}
 
=Normalized Mean Absolute Error=
 
Similarly, the Normalized Mean Absolute Error (NMAE) is given by  
Similarly, the Normalized Mean Absolute Error (NMAE) is given by  
(A5)
 
{{Equation|<math> NMAE = \frac{MAE}{range(x_m)}</math>|A5}}
 
The NMAE is often expressed in units of percent. Smaller values of NMAE indicate a better agreement between measured and calculated values.  
The NMAE is often expressed in units of percent. Smaller values of NMAE indicate a better agreement between measured and calculated values.  
Correlation Coefficient
 
=Correlation Coefficient=
 
Correlation is a measure of the strength and direction of a linear relation-ship between two variables. The correlation coefficient R is defined as  
Correlation is a measure of the strength and direction of a linear relation-ship between two variables. The correlation coefficient R is defined as  
(A5)
 
A correlation of 1 indicates a perfect one-to-one linear relationship and -1 indicates a negative relationship. The square of the correlation coefficient describes how much of the variance between two variables is described by a linear fit. The interpretation of the correlation coefficient depends on the context and purposes. For the present work, the following qualifications are used: 0.7<R2<1 = strong, 0.4<R2<0.7 = medium, 0.2<R2<0.4 = small, and R2<0.2 = none.
{{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_2 \rangle^2 }} </math>|A5}}
Bias
 
A correlation of 1 indicates a perfect one-to-one linear relationship and -1 indicates a negative relationship. The square of the correlation coefficient describes how much of the variance between two variables is described by a linear fit. The interpretation of the correlation coefficient depends on the context and purposes. For the present work, the following qualifications are used: 0.7<R<sup>2</sup><1 = strong, 0.4<R<sup>2</sup><0.7 = medium, 0.2<R<sup>2</sup><0.4 = small, and R<sup>2</sup><0.2 = none.
 
=Bias=
 
The Bias is defined as
The Bias is defined as
(A6)
 
Positive values indicate over-prediction and negative values indicate un-der-prediction.
{{Equation|<math> Bias = \langle x_c - x_m \rangle </math>|A6}}
 
Positive values indicate over-prediction and negative values indicate under-prediction.

Latest revision as of 20:38, 2 May 2015

15 Appendix E: Goodness of Fit Statistics

Brier Skill Score

The Brier Skill Score (BSS) is defined as

  (A1)

where the angled brackets indicate averaging, subscripts m, c, and 0 indi-cate measured, calculated, and initial values, respectively. 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. The following quantifications are used for de-scribing the BSS values: 0.8<BSS<1.0 = excellent, 0.6<BSS<0.8 = good, 0.3<BSS<0.6 = reasonable, 0<BSS<0.3 = poor, BSS<0 = bad.

Room Mean Squared Error

The Root Mean Squared Error (RMSE) is defined as

  (A2)

The RMSE has the same units as the measured data. Lower values of RMSE indicate a better match between measured and computed values.

Normalized Root Mean Squared Error

The Normalized Root Mean Squared Error (NRMSE) is

  (A3)

The NRMSE is often expressed in units of percent. The measured data range range(xm) can be estimated as max(xm) - min(xm) . Lower values of NRMSE indicate a better agreement between measured and computed values.

Mean Absolute Error

The Mean Absolute Error (MAE) is defined as

  (A4)

Normalized Mean Absolute Error

Similarly, the Normalized Mean Absolute Error (NMAE) is given by

  (A5)

The NMAE is often expressed in units of percent. Smaller values of NMAE indicate a better agreement between measured and calculated values.

Correlation Coefficient

Correlation is a measure of the strength and direction of a linear relation-ship between two variables. The correlation coefficient R is defined as

  (A5)

A correlation of 1 indicates a perfect one-to-one linear relationship and -1 indicates a negative relationship. The square of the correlation coefficient describes how much of the variance between two variables is described by a linear fit. The interpretation of the correlation coefficient depends on the context and purposes. For the present work, the following qualifications are used: 0.7<R2<1 = strong, 0.4<R2<0.7 = medium, 0.2<R2<0.4 = small, and R2<0.2 = none.

Bias

The Bias is defined as

  (A6)

Positive values indicate over-prediction and negative values indicate under-prediction.