We develop methods for analysing the `interaction' or dependence between points in a spatial point pattern, when the pattern is spatially inhomogeneous. Completely non-parametric study of interactions is possible using an analogue of the K-function. Alternatively one may assume a semi-parametric model in which a (parametrically specified) homogeneous Markov point process is subjected to (non-parametric) inhomogeneous independent thinning. The effectiveness of these approaches is tested on datasets representing the positions of trees in forests.

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spatstat is a package for analyzing spatial point pattern data. Its functionality includes exploratory data analysis, model-fitting, and simulation. It is designed to handle realistic datasets, including inhomogeneous point patterns, spatial sampling regions of arbitrary shape, extra covariate data, and ‘marks ’ attached to the points of the point pattern. A unique feature of spatstat is its generic algorithm for fitting point process models to point pattern data. The interface to this algorithm is a function ppm that is strongly analogous to lm and glm. This paper is a general description of spatstat and an introduction for new users.

[Read before The Royal Statistical Society at a meeting organized by the Research Section on

We consider the combination of path sampling and perfect simulation in the context of both likelihood inference and non-parametric Bayesian inference for pairwise interaction point processes. Several empirical results based on simulations and analysis of a dataset are presented, and the merits of using perfect simulation are discussed.

Summary. We describe practical techniques for fitting stochastic models to spatial point pattern data in the statistical package R. The techniques have been implemented in our package spatstat in R. They are demonstrated on two example datasets. 1

Introduction Spatial point processes play a fundamental role in spatial statistics. In the simplest case they model \small" objects that may be identied by a map of points showing stores, towns, plants, nests, galaxies or cases of a disease observed in a two or three dimensional region. The points may be decorated with marks (such as sizes or types) whereby marked point processes are obtained. The areas of applications are manifold: astronomy, geography, ecology, forestry, spatial epidemiology, image analysis, and many more. Currently spatial point processes is an active area of research, which probably will be of increasing importance for many new applications, as new technology such as geographical information systems makes huge amounts of spatial point process data available. Textbooks and review articles on dierent aspects of spatial point processes include Matheron (1975), Ripley (1977), Ripley (1981), Diggle (1983), Penttinen (1984), Daley &Vere-Jones (1988),

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We study a class of spatial point processes with a \directed" or \unilateral" Markov property, which we call Directed Markov Point Processes (DMPPs). These are the spatial-point-process counterparts of directed Markov random elds on a lattice. There are many conditional-independence properties that may be used to characterise spatial processes; one is the (F4) condition, which features in the theory of multi-dimensional-time-parameter counting processes. We show that the (F4) condition deserves to be called a Markov property, and explore its connection to the Markov conditional-independence property. The Mazziotto-Szpirglas exponential formula gives us an explicit formula for the likelihood of DMPPs. Connections between DMPPs and undirected spatial Markov point processes are explored, with speci c attention given to nite-range dependence.

In mining exploration it is often desired to predict the occurrence of ore deposits given other geological information, such as the locations of faults. More generally it is of interest to measure the spatial association between two spatial patterns observed in the same survey region. Berman (1986) developed parametric methods for conditional inference about a point process X given another spatial process Y . This paper proposes an alternative, nonparametric approach using distance methods, analogous to the use of the summary functions F , G and J for univariate point patterns. Our methods apply to a bivariate spatial process (X; Y ) consisting of a point process X and a random set Y . In particular we develop a bivariate analogue of the J-function of Van Lieshout and Baddeley (1997), which shows promise as a summary statistic, and turns out to be closely related to Berman's analysis. Properties of the bivariate J-function include a multiplicative identity under independent superposition, which has no analogue in the univariate case. Two geological examples are investigated. Keywords: empty space function, first contact distribution, geological lineaments, goodnessof -fit testing, J-function, line segment process, marked point process, Monte Carlo tests, Address for correspondence: Department of Mathematics & Statistics, University of Western Australia, Nedlands WA 6907, Australia. adrian@maths.uwa.edu.au ore deposits, Palm distribution, point processes, random sets, spatial statistics. AMS Mathematics Subject Classification (1995 Revision): Primary: 60D05. Secondary: 60G55, 62M30, 62G05. 2 1