# Model evaluation

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

\[\newcommand\NSE{\textrm{NSE}}\]

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.