Kling-Gupta linear regression

Abstract: Although the Kling-Gupta efficiency ($\mathrm{KGE}$) is widely adopted for model evaluation in hydrology, its properties as a statistical estimator remain unexplored. Investigating these properties is necessary because parameter estimation and forecast evaluation are inherently linked. To address this, we formalize the negatively oriented Kling-Gupta loss $L_\mathrm{KG} = (1 - \mathrm{KGE})^2$ within an extremum estimation framework (equivalent to maximizing $\mathrm{KGE}$) and analyze its behavior in multiple linear regression. We establish explicit formulas for the parameter estimates, showing that Kling-Gupta linear regression scales the ordinary least squares (OLS) coefficient vector by a variance-inflation factor governed by the sample variances and covariances of the predictors and the response. We show that Kling-Gupta linear regression predictions replicate the sample variance of the response on the training set, in contrast to the variance reduction inherent to OLS, while both estimators maintain the sample mean of the observations and achieve the same sample correlation between the predictions and the response. We show analytically that no single estimator can simultaneously maximize both the Nash-Sutcliffe efficiency $\mathrm{NSE}$ and $\mathrm{KGE}$: the OLS estimator attains the maximum possible $\mathrm{NSE}$ but not the maximum $\mathrm{KGE}$, while the Kling-Gupta estimator maximizes $\mathrm{KGE}$ at the cost of $\mathrm{NSE}$. We prove the almost sure convergence of the Kling-Gupta estimator to well-defined population limits and express those limits algebraically. Furthermore, we evaluate the training and test set performance metrics for both estimators, demonstrating that for each estimator the metrics on the training set and on an independent test set converge asymptotically to identical limits (though the limits differ between OLS and Kling-Gupta regression).