将复杂函数分解为可识别的凸凹成分，提升模型可解释性与预测精度。
Identifiable Convex-Concave Regression via Sub-gradient Regularised Least Squares
- 通过子梯度约束的仿射函数建模凸凹分量，实现结构化分解。
- 引入全局正交性约束，使残差与截距和输入变量无关，确保可识别性。
- 结合L1/L2正则化，提升泛化能力，适合政策评估与预测分析。
我们提出一种新型非参数回归方法——可识别凸凹非参数最小二乘法（ICCNLS），将目标函数分解为可加的形状约束分量，每个分量由子梯度受限的仿射函数表示。为解决凸凹分解中的仿射模糊性，引入全局统计正交性约束，确保残差与截距及输入变量不相关，从而实现分解可识别性并增强可解释性。进一步在子梯度上施加L1、L2及弹性网络正则化，以提升泛化性能并促进结构稀疏性。该方法在合成数据与真实世界数据（包括医疗定价数据）上进行评估，相比传统CNLS与凸凹差（DC）回归，展现出更优的预测准确率与模型简洁性。结果表明，统计可识别性结合凸凹结构与子梯度正则化，能构建适用于预测、基准评估与政策分析的可解释模型。
原文摘要 · Abstract (English)
We propose a novel nonparametric regression method that models complex input-output relationships as the sum of convex and concave components. The method-Identifiable Convex-Concave Nonparametric Least Squares (ICCNLS)-decomposes the target function into additive shape-constrained components, each represented via sub-gradient-constrained affine functions. To address the affine ambiguity inherent in convex-concave decompositions, we introduce global statistical orthogonality constraints, ensuring that residuals are uncorrelated with both intercept and input variables. This enforces decomposition identifiability and improves interpretability. We further incorporate L1, L2 and elastic net regularisation on sub-gradients to enhance generalisation and promote structural sparsity. The proposed method is evaluated on synthetic and real-world datasets, including healthcare pricing data, and demonstrates improved predictive accuracy and model simplicity compared to conventional CNLS and difference-of-convex (DC) regression approaches. Our results show that statistical identifiability, when paired with convex-concave structure and sub-gradient regularisation, yields interpretable models suited for forecasting, benchmarking, and policy evaluation.
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