发表机构
Bristol Myers Squibb; University of Pennsylvania（百时美施贵宝公司; 宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对空间域划分未知且响应与协变量关联随划分变化的非参数回归问题，提出联合学习划分与聚类回归函数的深度学习模型，经模拟和真实数据验证了其有效性。
AI 中文摘要
我们考虑当响应变量与协变量间的关联随空间域的未知划分而变化时的非参数回归问题。所提出的估计器联合学习划分和特定聚类的回归函数：仅依赖位置的神经网络确定聚类归属，而不同的神经网络描述各聚类内协变量与响应变量的关系。退火softmax松弛允许对原本离散的分配进行基于梯度的估计，同时采用图拉普拉斯和占用惩罚项以抑制碎片化区域和退化解。我们证明了在标签置换下的可识别性，在边界条件下对划分误差进行了界定，并将预测风险分解为回归和分配分量。当划分估计足够准确时，所得收敛速率与先验估计器一致。模拟结果表明，当回归曲面在空间边界处突变时，联合估计方法有效，包括存在非线性效应、区域大小不等、优先采样及空间相关误差的场景。最后，通过真实数据分析验证了所提方法的有效性。
英文摘要
We consider nonparametric regression when the association between a response and its covariates changes across an unknown partition of a spatial domain. The proposed estimator learns the partition and the cluster-specific regression functions jointly. A neural network depending only on location determines cluster membership, while separate neural networks describe the covariate--response relationship within the clusters. An annealed softmax relaxation permits gradient-based estimation of the otherwise discrete assignments. Graph-Laplacian and occupancy penalties are used to discourage fragmented regions and degenerate solutions. We establish identifiability up to label permutation, bound partition error under a margin condition, and decompose prediction risk into regression and assignment components. The resulting rate agrees with that of an oracle estimator when the partition is estimated sufficiently accurately. Simulations show that joint estimation is useful when regression surfaces change abruptly across spatial boundaries, including settings with nonlinear effects, unequal region sizes, preferential sampling, and spatially correlated errors. Finally, a real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.
