Residuals and Goodness-of-Fit Tests for Stationary Marked Gibbs Point Processes

Abstract

The inspection of residuals is a fundamental step to investigate the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed by Baddeley et al. as an empirical counterpart of the Campbell equilibrium equation for marked Gibbs point processes. The present paper focuses on stationary marked Gibbs point processes and deals with asymptotic properties of residuals for such processes. In particular, the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical ones (raw residuals, inverse residuals, Pearson residuals). Based on these asymptotic results, we define goodness-of-fit tests with Type-I error theoretically controlled. One of these tests constitutes an extension of the quadrat counting test widely used to test the null hypothesis of a homogeneous Poisson point process.

Almost sure behavior of functionals of stationary Gibbs point processes

“…Without additional assumption on X, Coeurjolly and Lavancier (2013) also proved that for any bounded domain W…”

“…Correctly renormalized, such functionals have been shown to converge almost surely to zero as the window of observation expands to R d . Coeurjolly and Lavancier (2013) have also established an asymptotic normality result. The goal of this paper is to go further in the study of such functionals by stating an almost sure rate of convergence.…”

“…Møller and Waagepetersen, 2004). Functionals of Gibbs point processes of the form L h − R h have been considered in Baddeley et al (2005Baddeley et al ( , 2008 and in Coeurjolly and Lavancier (2013). Correctly renormalized, such functionals have been shown to converge almost surely to zero as the window of observation expands to R d .…”

Calibration diagnostics for point process models via the probability integral transform

“…For marked point processes, on the other hand, alternative methods are called for. Schoenberg () considered thinned residuals for the space–time‐magnitude distribution of earthquake occurrences, and Coeurjolly and Lavancier () proposed an extension of the residual framework of Baddeley et al () that applies to stationary marked Gibbs processes. In our setting, marked point processes might be dealt with by applying the multivariate rank histogram proposed by Gneiting et al (), where the mark distribution is considered a separate component of a multivariate distribution.…”

Fast Covariance Estimation for Innovations Computed from a Spatial Gibbs Point Process

“…The h ‐innovation computed in a bounded domain Λ is the centred random variable defined by Baddeley et al () proposed to replace θ ⋆ in by a consistent estimator to obtain residuals for spatial point processes. Such residuals can be used as a diagnostic tool of goodness‐of‐fit and they have also been considered by Coeurjolly and Lavancier () and Baddeley et al () both from a theoretical and practical point of view.…”