Distributed Heterogeneity Learning for Generalized Partially Linear Models with Spatially Varying Coefficients¹

Spatial heterogeneity is of great importance in social, economic, and environmental science studies. The spatially varying coefficient model is a popular and effective spatial regression technique to address spatial heterogeneity. However, accounting for heterogeneity comes at the cost of reducing model parsimony. To balance flexibility and parsimony, this article develops a class of generalized partially linear spatially varying coefficient models which allow the inclusion of both constant and spatially varying effects of covariates. Another significant challenge in many applications comes from the enormous size of the spatial datasets collected from modern technologies. To tackle this challenge, we design a novel distributed heterogeneity learning (DHL) method based on bivariate spline smoothing over a triangulation of the domain. The proposed DHL algorithm has a simple, scalable, and communication-efficient implementation scheme that can almost achieve linear speedup. In addition, this article provides rigorous theoretical support for the DHL framework. We prove that the DHL constant coefficient estimators are asymptotic normal and the DHL spline estimators reach the same convergence rate as the global spline estimators obtained using the entire dataset. The proposed DHL method is evaluated through extensive simulation studies and analyses of U.S. loan application data.空间异质性（Spatial Heterogeneity）在社会科学、经济学与环境科学研究中具有重要意义。空间变系数模型（Spatially Varying Coefficient Model）是一类用于处理空间异质性的主流且高效的空间回归技术。然而，纳入异质性分析往往会以牺牲模型简约性为代价。为兼顾模型灵活性与简约性，本文提出了一类广义部分线性空间变系数模型，该模型可同时纳入协变量的恒定效应与空间变效应。在诸多实际应用中，另一项重大挑战源自现代技术采集的大规模空间数据集。为应对这一挑战，本文基于定义域三角剖分下的二元样条平滑方法，设计了一种全新的分布式异质性学习（Distributed Heterogeneity Learning，DHL）方法。所提出的DHL算法具备简洁、可扩展且通信高效的实现方案，可实现近似线性加速比。此外，本文为DHL框架提供了严谨的理论支撑。本文证明了DHL的恒定系数估计量服从渐近正态分布，且DHL样条估计量的收敛速率与使用全数据集得到的全局样条估计量一致。本文通过大量模拟实验与美国贷款申请数据分析，对所提出的DHL方法进行了验证。