Residual Analysis for Spatial Point Processes (with Discussion)

TLDR

Residuals for spatial point process models, based on conditional intensity, extend existing ad hoc model‑checking methods such as quadrat counts, scan statistics, kernel intensity estimates, and Berman’s diagnostic. The study defines residuals for spatial point process models and proposes diagnostic plots derived from them. The authors develop diagnostic tools by mapping spatial residuals to generalized linear model residuals, enabling the adaptation of GLM validation techniques to spatial point processes and yielding recommendations for diagnostic plots. Smoothed residual plots reveal spatial trend or covariate effects, while Q–Q‑plots detect interpoint interaction.

Abstract

Summary We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non-spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co-variate effects. Q–Q-plots of the residuals are effective in diagnosing interpoint interaction.