标题： 从协议指数中产生的损失函数 显示英文标题

标题： Loss functions arising from the index of agreement

摘要： 我们研究了协议指数损失函数$L_W$的理论性质，它是 Willmott 协议指数的负向对应物，是环境科学和工程中常用的度量标准。 我们证明了$L_W$位于 [0, 1] 范围内，具有平移和尺度不变性，并在拟合分布时估计参数$\Bbb{E}_{F}[\underline{y}] \pm \Bbb{V}_{F}^{1/2}[\underline{y}]$。 我们提出了$L_{\operatorname{NR}_2}$作为理论上的改进，它用欧几里得距离之和替换$L_W$的分母，更好地与底层几何直觉对齐。 这种新的损失函数保留了$L_W$的吸引人特性，同时还为线性模型参数估计提供了闭式解。 我们证明，当预测变量与因变量之间的相关性接近1时，来自平方误差、$L_{\operatorname{NR}_2}$和$L_W$的参数估计值会收敛。 这种行为在水文模型校准中得到体现（这是水资源工程中的核心任务），其中这些损失函数的性能几乎相同。 最后，我们建议对现有$L_p$-范数的协议指数变体进行潜在改进。 显示英文摘要

摘要： 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.