KGE (2009) is the original Kling-Gupta efficiency formulation of Gupta et al. (2009). It combines correlation, variability ratio, and mean ratio. The component definitions below use the same paired observations and predictions as the main score. Later KGE variants use different component definitions and are not implemented by this function.
Details
$$r=\frac{\sum_{i=1}^{n}(obs_i-\bar{obs})(pred_i-\bar{pred})} {\sqrt{\sum_{i=1}^{n}(obs_i-\bar{obs})^2\sum_{i=1}^{n}(pred_i-\bar{pred})^2}}, \quad \alpha=\sqrt{\frac{\sum_{i=1}^{n}(pred_i-\bar{pred})^2} {\sum_{i=1}^{n}(obs_i-\bar{obs})^2}}, \quad \beta=\frac{\bar{pred}}{\bar{obs}}.$$ $$\mathrm{KGE}_{2009}=1-\sqrt{(r-1)^2+(\alpha-1)^2+(\beta-1)^2}.$$
One is ideal. Values closer to one indicate agreement in linear association,
spread, and mean. The range is unbounded below and at most one. Zero is not
the observed-mean benchmark used for NSE. KGE (2009) is undefined when the
observed mean or either vector's standard deviation is zero, or fewer than
two pairs remain; it returns NA with a warning in those cases. As with NSE,
avoid treating KGE (2009) as the only measure of model quality; inspect its
components and complementary error metrics.
References
Gupta, H. V., Kling, H., Yilmaz, K. K., and Martinez, G. F. (2009). Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of Hydrology, 377, 80-91. doi:10.1016/j.jhydrol.2009.08.003