Kling-Gupta linear regression

Abstract

Kling-Gupta efficiency ($\mathrm{KGE}$) is a model performance evaluation metric widely used in hydrology, but its properties as a statistical estimator have remained unexplored. We formalize the Kling-Gupta loss $L_\mathrm{KG} = (1 - \mathrm{KGE})^2$ in an extremum estimation framework (maximizing $\mathrm{KGE}$) for multiple linear regression. We give explicit formulas showing that Kling-Gupta regression scales the ordinary least squares (OLS) coefficient vector by a variance-inflation factor depending on sample variances and covariances. Its predictions reproduce the training set response variance, unlike OLS's variance reduction, while both maintain the response mean and achieve the same sample correlation. We prove that no estimator simultaneously maximizes Nash-Sutcliffe efficiency ($\mathrm{NSE}$) and $\mathrm{KGE}$: OLS maximizes $\mathrm{NSE}$ but not $\mathrm{KGE}$, whereas Kling-Gupta regression maximizes $\mathrm{KGE}$ at the expense of $\mathrm{NSE}$. We establish almost-sure convergence of the Kling-Gupta estimator to well-defined population limits. The training and test set performance metrics for both estimators converge asymptotically to identical limits (different for OLS vs. Kling-Gupta). In a single-predictor model with fixed intercept, we identify conditions where a global minimum of $L_\mathrm{KG}$ does not exist because of discontinuity at zero slope. This work establishes a mathematical foundation for $\mathrm{KGE}$-based estimation and clarifies its effects on predictive performance in hydrologic modeling.