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Gibbs point processes for studying the development of spatial-temporal stochastic processes

Renshaw, E. and Sarkka, A. (2001) Gibbs point processes for studying the development of spatial-temporal stochastic processes. Computational Statistics and Data Analysis, 36 (1). pp. 85-105. ISSN 0167-9473

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Abstract

Although many studies of marked point processes analyse patterns in terms of purely spatial relationships, in real life spatial structure often develops dynamically through time. Here we use a specific space-time stochastic process to generate such patterns, with the aim of determining purely spatial summary measures from which we can infer underlying generating mechanisms of space-time stochastic processes. We use marked Gibbs processes in the estimation procedure, since these are commonly used models for point patterns with interactions, and can also be chosen to ensure that they possess similar interaction structure to the space-time processes under study. We restrict ourselves to Strauss-type pairwise interaction processes, as these are simple both to construct and interpret. Our analysis not only highlights the way in which Gibbs models are able to capture the interaction structure of the generating process, but it also demonstrates that very few statistical indicators are needed to determine the essence of the process. This contrasts markedly with the relatively large number of estimators usually needed to characterise a process in terms of spectral, autocorrelation or K-function representations. We show that the Strauss-type procedure is robust, i.e. it is not crucial to know the exact process-generating mechanism. Moreover, if we do possess additional information about the true mechanism, then the procedure becomes even more effective.