Score, Pseudo-Score and Residual Diagnostics for Spatial Point Process Models

TL;DR

This paper introduces new statistical tools for analyzing spatial point pattern data, including pseudo-score tests and residual diagnostics, supported by theoretical insights and software implementations.

Contribution

It generalizes the score test to a pseudo-score test based on pseudo-likelihood and develops diagnostics using point process residuals, enhancing model validation methods.

Findings

Supports the use of functional summary statistics like Ripley's K-function for testing spatial randomness.

Introduces the compensator of the K-function for testing fitted models.

Provides software for implementing the proposed diagnostics.

Abstract

We develop new tools for formal inference and informal model validation in the analysis of spatial point pattern data. The score test is generalized to a "pseudo-score" test derived from Besag's pseudo-likelihood, and to a class of diagnostics based on point process residuals. The results lend theoretical support to the established practice of using functional summary statistics, such as Ripley's $K$-function, when testing for complete spatial randomness; and they provide new tools such as the compensator of the $K$-function for testing other fitted models. The results also support localization methods such as the scan statistic and smoothed residual plots. Software for computing the diagnostics is provided.