# Nash–Sutcliffe model efficiency coefficient

The Nash–Sutcliffe model efficiency coefficient (NSE) is used to assess the predictive power of hydrological models. It is defined as:

${\displaystyle NSE=1-{\frac {\sum _{t=1}^{T}\left(Q_{m}^{t}-Q_{o}^{t}\right)^{2}}{\sum _{t=1}^{T}\left(Q_{o}^{t}-{\overline {Q}}_{o}\right)^{2}}}}$

where Qo is the mean of observed discharges, and Qm is modeled discharge. Qot is observed discharge at time t.[1]

Nash–Sutcliffe efficiency can range from −∞ to 1. An efficiency of 1 (NSE = 1) corresponds to a perfect match of modeled discharge to the observed data. An efficiency of 0 (NSE = 0) indicates that the model predictions are as accurate as the mean of the observed data, whereas an efficiency less than zero (NSE < 0) occurs when the observed mean is a better predictor than the model or, in other words, when the residual variance (described by the numerator in the expression above), is larger than the data variance (described by the denominator). Essentially, the closer the model efficiency is to 1, the more accurate the model is. Threshold values to indicate a model of sufficient quality have been suggested between 0.5 < NSE < 0.65.[2][3] For the case of regression procedures (i.e. when the total sum of squares can be partitioned into error and regression components), the Nash–Sutcliffe efficiency is equivalent to the coefficient of determination (R2), thus ranging between 0 and 1.

The efficiency coefficient is sensitive to extreme values and might yield sub-optimal results when the dataset contains large outliers in it. To address this a modified version of NSE has been suggested where the sum of squares in the denominator of NSE is raised to 1 instead of 2 and the resulting modified NSE values compared to the original NSE values to assess the potential effect of extreme values.[4] A test significance for NSE to assess its robustness has been proposed whereby the model can be objectively accepted or rejected based on the probability value of obtaining NSE > threshold (0.65 or other selected by the user).[2]

Nash–Sutcliffe efficiency can be used to quantitatively describe the accuracy of model outputs other than discharge. This indicator can be used to describe the predictive accuracy of other models as long as there is observed data to compare the model results to. For example, Nash–Sutcliffe efficiency has been reported in scientific literature for model simulations of discharge; water quality constituents such as sediment, nitrogen, and phosphorus loading.[3] Other applications are the use of Nash–Sutcliffe coefficients to optimize parameter values of geophysical models, such as models to simulate the coupling between isotope behavior and soil evolution.[5]