Kling–Gupta efficiency

The Kling–Gupta efficiency (KGE) is a goodness-of-fit indicator widely used in the hydrologic sciences for comparing simulations to observations. It was created by hydrologic scientists Harald Kling and Hoshin Vijai Gupta.^[1] Its creators intended for it to improve upon widely used metrics such as the coefficient of determination and the Nash–Sutcliffe model efficiency coefficient.

${\displaystyle {\text{KGE}}=1-{\sqrt {(r-1)^{2}+(\alpha -1)^{2}+(\beta -1)^{2}}}}$

where:

• ${\textstyle r}$ is the Pearson correlation coefficient,
• ${\textstyle \alpha }$ is a term representing the variability of prediction errors,
• ${\textstyle \beta }$ is a bias term.

The terms ${\textstyle \alpha }$ and ${\textstyle \beta }$ are defined as follows:

${\displaystyle \beta ={\frac {\mu _{s}}{\mu _{o}}}}$

where:

• ${\textstyle \mu _{s}}$ is the mean of the simulated time series (e.g.: flows predicted by the model)
• ${\textstyle \mu _{o}}$ is the mean of the observed time series

and

${\displaystyle \alpha ={\frac {\sigma _{s}}{\sigma _{o}}}}$

where:

• ${\textstyle \sigma _{s}^{2}}$ is the variance of the simulated time series, so ${\textstyle \sigma _{s}}$ is estimated by the standard deviation of simulated data.
• ${\textstyle \sigma _{o}^{2}}$ is the variance of the observed time series

A modified version, KGE', was proposed by Kling et al. in 2012.^[2]