改进留一法交叉验证，提升函数逼近误差估计精度
Weighted Leave-One-Out Cross Validation
- 基于高斯过程构建加权留一法误差估计器
- 相比传统方法，积分平方误差估计更精确
- 适合需要精准误差评估的模型选择场景
我们提出一种加权留一法交叉验证方法，用于估计在有限采样点上依赖线性预测器逼近未知函数时的积分平方误差（ISE）。该方法基于高斯过程假设，构造任意未采样点处平方预测误差的最佳线性估计，利用平方留一法残差。通过理论分析和数值实验，验证了该估计器对高斯过程核函数选择具有鲁棒性。结果表明，与传统未加权留一法相比，ISE估计精度显著提升。文中还通过实例简要探讨了其在模型选择中的应用。
原文摘要 · Abstract (English)
We present a weighted version of Leave-One-Out (LOO) cross-validation for estimating the Integrated Squared Error (ISE) when approximating an unknown function by a predictor that depends linearly on evaluations of the function over a finite collection of sites. The method relies on the construction of the best linear estimator of the squared prediction error at an arbitrary unsampled site based on squared LOO residuals, assuming that the function is a realization of a Gaussian Process (GP). A theoretical analysis of performance of the ISE estimator is presented, and robustness with respect to the choice of the GP kernel is investigated first analytically, then through numerical examples. Overall, the estimation of ISE is significantly more precise than with classical, unweighted, LOO cross validation. Application to model selection is briefly considered through examples.
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