用可微分聚类分配训练神经网络，自动发现空间分区并建模区域差异。
A Deep Learning Model for Spatially Clustered Data via Differentiable Cluster Assignment
- 通过位置相关神经网络动态分配空间聚类，支持联合优化分区与回归函数。
- 在边界突变、非线性效应等复杂场景下，预测误差显著低于传统方法。
- 适合处理空间异质性强、分区未知的地理统计或环境数据建模任务。
当响应变量与协变量的关系在未知的空间划分上发生变化时，本文研究非参数回归问题。提出的估计器联合学习空间划分与各聚类内的回归函数：仅依赖位置的神经网络决定聚类归属，而独立神经网络刻画聚类内协变量-响应关系。采用退火软最大化松弛实现离散分配的梯度估计。引入图拉普拉斯和占用惩罚项以避免区域碎片化和退化解。理论方面，证明了在标签置换意义下的可识别性，在边缘条件下界定了划分误差，并将预测风险分解为回归与分配两部分。当分区估计足够准确时，收敛速率与已知分区的最优估计器一致。模拟结果显示，在回归面突变、非线性效应、区域大小不均、偏好采样及空间相关误差等情形下，联合估计显著提升性能。最后，真实数据分析验证了该方法的有效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。