# Model evaluation

## 1 difference between Nash-Sutcliffe efficiency and coefficient of determination

This response discusses the major difference between the coefficient of determination (\(R^2\)) and the Nash-Sutcliffe coefficient (\(NSE\)). These two coefficients are defined, respectively, as follows: $$R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}$$ and $$NSE=1-\frac{\sum(y_i-y_{i,\text{sim}})^2}{\sum(y_i-\bar{y})^2}$$ where \(y_i\) is the observed value of the variable of interest \(y\), \(\bar{y}\) is the mean of \(y_i\), \(\hat{y}_i\) is the predicted value from the statistical model inferred from the observed values, and \(y_{i,\text{sim}}\) is the predicted value from a simulation model. The only difference in these two definitions is \(\hat{y}_i\) versus \(y_{i,\text{sim}}\). That is, \(R^2\) describes how far individual observations are from the statistical model built from those values while \(NSE\) evaluates how far individual observations are from simulated predictions. Since \(y_{i,\text{sim}}\) is not derived from observations, the major difference is that \(R^2\) describes observed data only, but \(NSE\) describes both observed data and simulated predictions.