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The Nash-Sutcliffe Coefficient (E) is commonly used to assess the predictive power of a model. It is defined as
The Nash-Sutcliffe Coefficient (E) is commonly used to assess the predictive power of a model. It is defined as
{{Equation|<math> E = 1 -  \frac{\bigg\langle \big(x_m-x_c\big)^2  \bigg\rangle}{\bigg\langle  \big(x_m-  \langle x_m \rangle \big)^2 \bigg\rangle }  </math>|2=2}}
{{Equation|<math> E = 1 -  \frac{\bigg\langle \big(x_m-x_c\big)^2  \bigg\rangle}{\bigg\langle  \big(x_m-  \langle x_m \rangle \big)^2 \bigg\rangle }  </math>|2=2}}
where where <math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values. The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.
where <math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values. The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.


== Root-Mean-Squared Error ==
== Root-Mean-Squared Error ==
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== Normalized-Root-Mean-Squared Error ==
== Normalized-Root-Mean-Squared Error ==
In order to make comparing different RMSE with different units or scales (lab vs field) several non-dimensional forms of the RMSE have been proposed in literature. Here the Normalized-Root-Mean-Squared Error (NRMSE) is defined as
In order to make comparing different RMSE with different units or scales (lab vs field) several non-dimensional forms of the RMSE have been proposed in literature. Here the Normalized-Root-Mean-Squared Error (NRMSE) is defined as
{{Equation|<math> NRMSE = \frac{RMSE}{\text{Range}(x_m)} </math>|2=4}}
{{Equation|<math> NRMSE = \frac{\sqrt{ \bigg\langle \big( x_m - x_c  \big)^2  \bigg\rangle  }}{\max{(x_m)}-\min{(x_m)}} </math>|2=4}}
where <math>\text{Range}(x_m) = \max{(x_m)}-\min{(x_m)} </math> is the range of the data. The NRMSE is often expressed in units of percent. Smaller values indicate a better agreement between measured and calculated values.
where <math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values. The NRMSE is often expressed in units of percent. Smaller values indicate a better agreement between measured and calculated values.


== Mean-Absolute Error ==
== Mean-Absolute Error ==
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== Normalized-Mean-Absolute Error ==
== Normalized-Mean-Absolute Error ==
The normalized-Mean-Absolute Error is defined as  
The normalized-Mean-Absolute Error is defined as  
{{Equation|<math>  NMAE = \frac{MAE}{ \text{Range}(x_m) }  </math>|2=6}}
{{Equation|<math>  NMAE = \frac{MAE}{ \max{(x_m)}-\min{(x_m)} }  </math>|2=6}}
where where <math>\text{Range}(x_m) = \max{(x_m)}-\min{(x_m)}, </math> is the range of the data.<math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values.
where <math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values.


== Correlation coefficient is defined as ==
== Correlation coefficient is defined as ==
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The bias is a measure of the over or under estimation and is defined as  
The bias is a measure of the over or under estimation and is defined as  
{{Equation|<math>  B =  \langle x_c - x_m \rangle  </math>|2=8}}
{{Equation|<math>  B =  \langle x_c - x_m \rangle  </math>|2=8}}
 
where <math>x_m</math> is the measured or observed values, <math>x_c</math> is the calculated values. The bias is a measure of the over or under prediction of a variable. Positive values indicate overprediction and negative values indicate underprediction.
The bias is a measure of the over or under prediction of a variable. Positive values indicate overprediction and negative values indicate underprediction.


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[[CMS#Documentation_Portal | Documentation Portal]]

Revision as of 17:19, 6 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

The Bier Skill Score (BSS) is given by

  (1)

where is the measured or observed values, is the calculated values and is the initial measured values. 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.

Nash-Sutcliffe Coefficient

The Nash-Sutcliffe Coefficient (E) is commonly used to assess the predictive power of a model. It is defined as

  (2)

where is the measured or observed values, is the calculated values. The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.

Root-Mean-Squared Error

The Root-Mean-Squared Error (RMSE) also referred to as Root-Mean-Squared Deviation (RMSD) is defined as

  (3)

where where is the measured or observed values, is the calculated values. The RMSE has the same units as the measured and calculated data.

Normalized-Root-Mean-Squared Error

In order to make comparing different RMSE with different units or scales (lab vs field) several non-dimensional forms of the RMSE have been proposed in literature. Here the Normalized-Root-Mean-Squared Error (NRMSE) is defined as

  (4)

where is the measured or observed values, is the calculated values. The NRMSE is often expressed in units of percent. Smaller values indicate a better agreement between measured and calculated values.

Mean-Absolute Error

  (5)

where where is the measured or observed values, is the calculated values.

Normalized-Mean-Absolute Error

The normalized-Mean-Absolute Error is defined as

  (6)

where is the measured or observed values, is the calculated values.

Correlation coefficient is defined as

Correlation is a measure of the strength and direction of a linear relationship between two variables. The correlation coefficient is defined as

  (7)

where where is the measured or observed values, is the calculated values. 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.

Bias

The bias is a measure of the over or under estimation and is defined as

  (8)

where is the measured or observed values, is the calculated values. The bias is a measure of the over or under prediction of a variable. Positive values indicate overprediction and negative values indicate underprediction.


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