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A nonparametric measure of spatial interaction in point patterns. (English) Zbl 0898.62118
Summary: The strength and range of interpoint interactions in a spatial point process can be quantified by the function \(J= (1-G)/(1-F)\), where \(G\) is the nearest-neighbour distance distribution function and \(F\) the empty space function of the process. \(J(r)\) is identically equal to 1 for a Poisson process; values of \(J(r)\) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, \(J(r)\) is constant for distances \(r\) greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from \(J\) without parametric model assumptions. It is also possible to evaluate \(J(r)\) explicitly for many point process models, so that \(J\) is also useful for parameter estimation.
Various properties are derived, including the fact that the \(J\) function of the superposition of independent point processes is a weighted mean of the \(J\) functions of the individual processes. Estimators of \(J\) can be constructed from standard estimators of \(F\) and \(G\). We compute estimates of \(J\) for several standard point pattern datasets and implement a Monte Carlo test for complete spatial randomness.

MSC:
62M30 Inference from spatial processes
62G05 Nonparametric estimation
Software:
spatial
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