Local signal detection on irregular domains with generalized varying coefficient models.

Area of Science:

• Spatial analysis
• Statistical modeling
• Geostatistics

Background:

• Spatial analysis requires quantifying heterogeneity.
• Generalized Spatially Varying Coefficient Models (GSVCM) address spatial heterogeneity by allowing coefficients to vary.
• Detecting local signals within these models is crucial.

Purpose of the Study:

• To propose a penalized bivariate spline method for detecting local signals in GSVCM.
• To develop confidence regions for quantifying uncertainty in estimated null regions.
• To establish the consistency of the proposed nonparametric coefficient function and null region estimation.

Main Methods:

• Utilizing bivariate splines on triangulations to approximate nonparametric varying coefficient functions.
• Applying a local penalty on L2 norms of spline coefficients per triangle to identify null regions.
• Developing an efficient algorithm using local quadratic approximation for estimation.

Main Results:

• The method effectively detects local signals and identifies regions of zero effects in GSVCM.
• Confidence regions provide uncertainty quantification for estimated null regions.
• Consistency of estimated nonparametric coefficient functions and null regions is established.

Conclusions:

• The penalized bivariate spline method offers a robust approach for analyzing spatial heterogeneity using GSVCM.
• The proposed technique efficiently handles irregular domains and provides reliable inference.
• Numerical evaluations demonstrate the method's performance in simulations and real-world data.