Kernel Regression for Spatial and Spatio-Temporal Residual Risk: Application to School Shootings in the Contiguous United States
School gun violence in the United States is a complex phenomenon spanning social, epidemiological, demographic, and political dimensions. It remains unclear where incidents are unusually concentrated nationally after accounting for the distribution and characteristics of schools. Using a newly linked case-control dataset comprising 959 gun-violence incidents at public K-12 schools in the contiguous United States during 2000-2024, we develop a semiparametric kernel-regression framework combining school-level predictors with spatial and continuously evolving spatio-temporal residual structure. Fisher-weighted orthogonalisation defines how predictor-aligned variation is allocated between fixed and smooth components, while repeated control sampling and Monte Carlo reassignment support stable mapping and local exceedance assessment. The models identify stable school-level associations, including substantially higher adjusted odds for larger, middle, and high schools, while revealing residual structure beyond the background distribution of schools. Elevated residual odds become concentrated in a broad central-eastern corridor from the mid-2010s onward, with the strongest evidence in recent years. The analysis offers both statistical and application-specific insights. Statistically, it shows how covariate-adjusted residual surfaces can characterise local departures in case-control processes evolving over space and time. For the application, it provides epidemiological clues identifying regions in which broader social, policy, and environmental conditions may warrant targeted investigation.