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arxiv: 2606.09391 · v1 · pith:XRD6HBS4new · submitted 2026-06-08 · 🧮 math.ST · physics.ao-ph· stat.ME· stat.TH

Kling-Gupta linear regression

Hristos Tyralis , Georgia Papacharalampous This is my paper

Pith reviewed 2026-06-27 14:47 UTC · model grok-4.3

classification 🧮 math.ST physics.ao-phstat.MEstat.TH
keywords Kling-Gupta efficiencylinear regressionparameter estimationNash-Sutcliffe efficiencyextremum estimationvariance inflation factorhydrology modeling
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The pith

Kling-Gupta linear regression scales the ordinary least squares coefficient vector by a variance-inflation factor based on sample variances and covariances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper formalizes minimization of the Kling-Gupta loss within multiple linear regression and derives explicit formulas for the resulting parameter estimates. The estimates equal the ordinary least squares solution multiplied by a scaling factor that depends on the sample variances of the predictors and response along with the relevant covariances. A sympathetic reader would care because KGE is a standard evaluation metric in hydrology, so using it directly for fitting links estimation and assessment in a way that preserves the observed response variance exactly. The work further shows that no single estimator can maximize both the Nash-Sutcliffe efficiency and the Kling-Gupta efficiency simultaneously, and that the Kling-Gupta estimator converges almost surely to explicit population limits.

Core claim

Minimizing the negatively oriented Kling-Gupta loss L_KG = (1 - KGE)^2 in multiple linear regression produces coefficient estimates that scale the ordinary least squares vector by a variance-inflation factor governed by the sample variances and covariances of the predictors and response. The resulting predictions replicate the sample variance of the observations on the training set, while both the Kling-Gupta and ordinary least squares estimators match the sample mean of the observations and achieve identical sample correlations between predictions and observations. The Kling-Gupta estimator converges almost surely to well-defined population limits expressed algebraically in terms of the und

What carries the argument

The explicit scaling of the ordinary least squares coefficient vector by a variance-inflation factor determined by sample variances and covariances.

If this is right

  • Kling-Gupta regression predictions exactly replicate the sample variance of the response on the training set.
  • Both estimators match the sample mean of the observations and achieve the same sample correlation between predictions and observations.
  • The ordinary least squares estimator attains the maximum possible Nash-Sutcliffe efficiency but not the maximum Kling-Gupta efficiency, while the Kling-Gupta estimator does the reverse.
  • The Kling-Gupta estimator converges almost surely to explicit algebraic population limits, and training and test performance metrics converge to identical asymptotic values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In hydrology applications, fitting directly to the Kling-Gupta loss may produce models whose predictions better retain the variability seen in observed data.
  • The demonstrated impossibility of jointly maximizing NSE and KGE creates a concrete trade-off that modelers must navigate when selecting a loss function.
  • The scaling relation derived here could be tested for persistence when the Kling-Gupta loss is applied inside nonlinear or regularized regression frameworks.

Load-bearing premise

The negatively oriented Kling-Gupta loss can be directly minimized in a standard multiple linear regression model using the usual sample moments without additional constraints or modifications to the loss definition.

What would settle it

On any finite dataset compute the ordinary least squares coefficients, the sample variances and covariances, and the stated variance-inflation factor; if the Kling-Gupta coefficient estimates deviate from the predicted scaled vector then the explicit formulas are falsified.

Figures

Figures reproduced from arXiv: 2606.09391 by Georgia Papacharalampous, Hristos Tyralis.

Figure 1
Figure 1. Figure 1: Visual comparison of OLS (in blue) and Kling-Gupta linear regression (in red) [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Asymptotic convergence of performance metrics as a function of sample size for [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Asymptotic performance metrics (NSE and KGE) from Table [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kling-Gupta loss LKG(axn + b01n, yn ) as a function of a for a single-predictor linear model with sample statistics µ(xn) = 2, µ(yn ) = 1, σ(xn) = 1, σ(yn ) = 1, and ρ(xn, yn ) = −1: (a) b0 = −0.8, and (b) b0 = 0.5. The positive local optimizer a+ is given by eq. (B.99). Open black circles indicate the points where the Kling-Gupta loss is undefined. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scatterplot comparing observed and predicted streamflow for catchment [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time series plot of observed streamflow and predictions generated by the four [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance comparison of the OLS and Kling-Gupta linear regression models [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Decomposition of the Kling-Gupta loss into its three components for the OLS [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
read the original 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper formalizes the negatively oriented Kling-Gupta loss L_KG = (1 - KGE)^2 within an extremum estimation framework for multiple linear regression. It derives explicit algebraic formulas for the parameter estimates, showing that the Kling-Gupta estimator scales the OLS coefficient vector by a variance-inflation factor based on sample variances and covariances. The resulting predictions match the sample variance of the response (unlike OLS), while both estimators match the sample mean and achieve identical sample correlation with the response. The paper proves that no single estimator can simultaneously maximize NSE and KGE, establishes almost-sure convergence of the Kling-Gupta estimator and its metrics to explicit population limits via the LLN, and shows that training and test metrics converge to the same asymptotic limits (differing between the two estimators).

Significance. If the central derivations hold, the manuscript supplies explicit closed-form expressions, a variance-matching property, an NSE-KGE incompatibility result, and LLN-based convergence statements for KGE-based regression. These algebraic characterizations and the explicit population limits constitute clear strengths for a statistics paper, providing a precise link between a widely used hydrological metric and standard regression estimators.

minor comments (2)
  1. [§3] §3: the variance-inflation factor is described in the text but would benefit from an explicit numbered equation to facilitate cross-reference with the population-limit expressions later in the paper.
  2. [Notation] Notation section: sample moments and population quantities are used throughout; a brief table or sentence distinguishing the two (e.g., r vs. ρ) would improve readability of the convergence statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of the manuscript, the recognition of its algebraic characterizations and population limits as strengths, and the recommendation for minor revision. We are pleased that the central results on the Kling-Gupta estimator, its variance-matching property, the NSE-KGE incompatibility, and the LLN convergence are viewed favorably.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper starts from the explicit definition of the negatively oriented KGE loss L_KG = (1 - KGE)^2 and standard sample moments (means, variances, covariances, correlation) within an extremum estimation framework for multiple linear regression. It derives the closed-form estimator as a scaled OLS coefficient vector by maximizing correlation subject to the variance-matching constraint that is built into KGE, then applies the LLN to obtain population limits expressed algebraically in the same moments. These steps rely only on the algebraic properties of sample correlation and variance plus standard convergence arguments; no parameter is fitted to a subset and then renamed as a prediction, no self-citation supplies a uniqueness theorem, and no ansatz is smuggled in. The incompatibility between NSE and KGE maximizers follows directly from the differing objective functions. The analysis is therefore independent of its inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical properties of linear regression moments and the pre-existing definition of KGE from hydrology; no new free parameters are introduced and no entities are postulated.

axioms (2)
  • standard math Sample variances and covariances exist and the predictor matrix permits the usual OLS inversion
    Required to define the variance-inflation factor and explicit coefficient formulas.
  • domain assumption KGE is composed of its three standard components (correlation, bias ratio, variability ratio) as defined in the hydrology literature
    The loss function L_KG is constructed directly from this definition.

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