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 paper 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.在空间分析领域，理解并量化空间或时间异质性是一项核心任务。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发生变化，以适配空间异质性的高效分析框架。我们提出了一种惩罚化二元样条方法，用于在广义空间变系数模型中检测局部信号。其核心思路为：利用三角剖分上定义的二元样条近似非参数变系数函数，并对每个三角单元的样条系数的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零效应空区域的不确定性。我们的方法通过三角剖分对感兴趣区域进行划分，并可高效近似不规则空间域。除此之外，我们提出了一种基于局部二次近似的高效算法，用于求解所提估计量。我们还证明了所估计的非参数系数函数以及零效应空区域的一致性。本文通过仿真研究与实际数据分析两种场景，对所提方法的数值性能进行了评估。