Residual Analysis for Spatial Point Processes (with Discussion)

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 covariate effects. Q-Q-plots of the residuals are effective in diagnosing interpoint interaction.

Prospective and retrospective evaluation of five-year earthquake forecast models for California

“…Residuals scores from the point process modelling literature (Lawson 1993;Baddeley et al 2005;Clements et al 2011) are well-suited to resolve this particular issue. Forecasts can be represented as space-time point process models, where binwise forecasts are represented as infinitesimal seismicity rates at specific locations.…”

Central limit theorems for empirical product densities of stationary point processes

“…. , f k ) is decomposable with respect to the decomposition T = {T 1 , T 2 } if there exist a decomposition P (1) of {1, . .…”

“…Baddeley et al [1], Cressie [3], Diggle [6], Illian et al [10], and Stoyan et al [16]. Compared with the pair correlation function, the second-order product density contains additional information if the stationary point process is anisotropic.…”

Testing long‐term earthquake forecasts: Likelihood methods and error diagrams

“…Baddeley et al (2005), Baddeley (2007), Vere-Jones (2003, 2008) provide reviews. As we explain below, the earthquake forecasting problem is different in many respects from regular density evaluation and requires special treatment.…”