改进局部多项式回归的加权机制，提升对异常值的鲁棒性。
Robust Local Polynomial Regression with Similarity Kernels
- 用预测变量和响应变量共同构造相似性核函数，动态调整权重。
- 在真实数据上实验显示，偏差低于迭代稳健LOWESS方法。
- 适合处理含异常值的复杂非线性关系建模，如金融或生物数据分析。
局部多项式回归（LPR）因其灵活性和简便性被广泛用于建模复杂关系，通过在局部数据子集上拟合低阶多项式并按距离加权来估计回归函数。然而，传统LPR对异常值和高杠杆点敏感，严重影响估计精度。本文重新审视用于计算回归权重的核函数，提出一种新框架，将预测变量与响应变量同时纳入加权机制。核心思想是采用条件密度核，通过局部密度估计抑制异常值的影响。该方法已用Python实现并公开于https://github.com/yaniv-shulman/rsklpr。群体分析量化了基于密度的鲁棒加权所引入的偏倚，实验表明其经验偏倚低于迭代稳健LOWESS，同时保持与标准LOWESS相当的性能。这一进展为传统LPR提供了有前景的扩展，推动了鲁棒回归在实际场景中的应用。
原文摘要 · Abstract (English)
Local Polynomial Regression (LPR) is a widely used nonparametric method for modeling complex relationships due to its flexibility and simplicity. It estimates a regression function by fitting low-degree polynomials to localized subsets of the data, weighted by proximity. However, traditional LPR is sensitive to outliers and high-leverage points, which can significantly affect estimation accuracy. This paper revisits the kernel function used to compute regression weights and proposes a novel framework that incorporates both predictor and response variables in the weighting mechanism. The focus of this work is a conditional density kernel that robustly estimates weights by mitigating the influence of outliers through localized density estimation. The proposed method is implemented in Python and is publicly available at https://github.com/yaniv-shulman/rsklpr. The population analysis quantifies the bias induced by density-based robust weighting, and the reported experiments show lower empirical bias than iterative robust LOWESS while remaining competitive with standard LOWESS. This advancement provides a promising extension to traditional LPR, opening new possibilities for robust regression applications.
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