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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msub> </mml:math> 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.