Local Signal Detection on Irregular Domains with Generalized Varying Coefficient Models

In spatial analysis, it is essential to understand and quantify spatial or temporal heterogeneity. This article focuses on the generalized spatially varying coefficient model (GSVCM), a powerful framework to accommodate spatial heterogeneity by allowing regression coefficients to vary in a given spatial domain. We propose a penalized bivariate spline method for detecting local signals in GSVCM. The key idea is to use bivariate splines defined on triangulation to approximate nonparametric varying coefficient functions and impose a local penalty on L2 norms of spline coefficients for each triangle to identify null regions of zero effects. Moreover, we develop model confidence regions as the inference tool to quantify the uncertainty of the estimated null regions. Our method partitions the region of interest using triangulation and efficiently approximates irregular domains. In addition, we propose an efficient algorithm to obtain the proposed estimator using the local quadratic approximation. We also establish the consistency of estimated nonparametric coefficient functions and the estimated null regions. The numerical performance of the proposed method is evaluated in both simulation cases and real data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.在空间分析中，认知并量化空间或时间异质性是一项至关重要的工作。本文聚焦于广义空间变系数模型（generalized spatially varying coefficient model, GSVCM），这是一种通过允许回归系数在指定空间域内变化，以适配空间异质性的高效分析框架。我们提出了一种用于广义空间变系数模型局部信号检测的惩罚型二元样条方法，其核心思路是利用三角剖分上定义的二元样条近似非参数变系数函数，并针对每个三角形的样条系数的L2范数施加局部惩罚，以识别效应为零的空区域。此外，我们构建了模型置信区域作为推断工具，用于量化估计得到的空区域的不确定性。我们的方法通过三角剖分对感兴趣区域进行划分，可高效近似不规则空间域。另外，我们基于局部二次近似方法，提出了一种可高效求解所提估计量的算法。我们还证明了所估计的非参数系数函数以及空区域估计结果的一致性。我们通过模拟研究与实际数据分析两类场景，对所提方法的数值性能进行了评估。本文的补充材料可在线获取，其中包含可用于复现研究成果的材料的标准化说明。