We examine the theoretical properties of the index of agreement loss function $L_W$, the negatively oriented counterpart of Willmott's index of agreement, a common metric in environmental sciences and engineering. We prove that $L_W$ is bounded within [0, 1], translation and scale invariant, and estimates the parameter $\Bbb{E}_{F}[\underline{y}] \pm \Bbb{V}_{F}^{1/2}[\underline{y}]$ when fitting a distribution. We propose $L_{\operatorname{NR}_2}$ as a theoretical improvement, which replaces the denominator of $L_W$ with the sum of Euclidean distances, better aligning with the underlying geometric intuition. This new loss function retains the appealing properties of $L_W$ but also admits closed-form solutions for linear model parameter estimation. We show that as the correlation between predictors and the dependent variable approaches 1, parameter estimates from squared error, $L_{\operatorname{NR}_2}$ and $L_W$ converge. This behavior is mirrored in hydrologic model calibration (a core task in water resources engineering), where performance becomes nearly identical across these loss functions. Finally, we suggest potential improvements for existing $L_p$-norm variants of the index of agreement.

翻译：本文研究了Willmott一致性指数的负向形式——一致性指数损失函数$L_W$的理论性质，该指标在环境科学与工程领域应用广泛。我们证明$L_W$具有[0, 1]的有界性、平移不变性与尺度不变性，且在拟合分布时估计的参数为$\Bbb{E}_{F}[\underline{y}] \pm \Bbb{V}_{F}^{1/2}[\underline{y}]$。我们提出理论改进版本$L_{\operatorname{NR}_2}$，将$L_W$的分母替换为欧几里得距离之和，从而更契合其几何直观。这一新型损失函数在保留$L_W$优良特性的同时，还能为线性模型参数估计提供闭式解。我们证明当预测变量与因变量的相关性趋近于1时，平方误差损失、$L_{\operatorname{NR}_2}$与$L_W$所得参数估计将收敛。这种特性在水文模型率定（水资源工程的核心任务）中得到印证——不同损失函数在该场景下表现出近乎一致的性能。最后，我们对现有$L_p$范数变体的一致性指数提出了可能的改进方向。