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Non and semiparametric estimation of interaction in inhomogeneous p oint patterns y z Adrian Baddeley Jesp er Mller Rasmus Waagep etersen Abstract We develop metho ds for analysing the interaction or dep endence b etween p oints in a spatial p oint pattern when the pattern is spatially inhomogeneous Completely nonparametric study of interactions is p ossible using an analogue of the K function Alternatively one may assume a semiparametric mo del in which a parametrically sp ecied homogeneous Markov p oint pro cess is sub jected to nonparametric inho mogeneous indep endent thinning The eectiveness of these approaches is tested on datasets representing the p ositions of trees in forests Keywords Kernel estimation Markov chain Monte Carlo Markov p oint pro cess Second order metho ds Semiparametric inference Spatial statistics Summary statistics Thin ning Intro duction In the analysis of spatial p oint patterns one of the main ob jectives is often to assess the strength of interaction or dep endence b etween the p oints Metho ds for investigat ing interp oint interaction include nonparametric summaries such as the second moment function K Ripley Diggle and explicit parametric mo dels such as Markov p oint pro cesses Ripley and Kelly Baddeley and Mller However the exist ing nonparametric summaries rely on the assumption that the p oint pattern is spatially homogeneous stationary andor isotropic If an initial analysis suggests that the pattern has nonconstant intensity then current nonparametric techniqu