Abstract
The intensity of a Gibbs point process model is usually an intractable function of the model parameters. This is a severe restriction on the practical application of such models. We develop a new approximation for the intensity of a stationary Gibbs point process on $\mathbb{R}^{d}$. For pairwise interaction processes, the approximation can be computed rapidly and is surprisingly accurate. The new approximation is qualitatively similar to the mean field approximation, but is far more accurate, and does not exhibit the same pathologies. It may be regarded as a counterpart of the Percus-Yevick approximation.
Highlights
In the statistical analysis of spatial point pattern data, an important role is played by Gibbs point process models, especially pairwise interaction processes [12, 45, 31]
Many important properties of these models are intractable, including the intensity, higher moments, and the partition function. This intractability is a severe impediment to the use of Gibbs models in applied probability and statistics, and it motivated the invention of Markov Chain Monte Carlo methods [29]
We proposed an approximation to the intensity of a Gibbs point process
Summary
In the statistical analysis of spatial point pattern data, an important role is played by Gibbs point process models, especially pairwise interaction processes [12, 45, 31]. Many important properties of these models are intractable, including the intensity (expected number of points per unit volume), higher moments, and the partition function (normalising constant of the likelihood) This intractability is a severe impediment to the use of Gibbs models in applied probability and statistics, and it motivated the invention of Markov Chain Monte Carlo methods [29]. It does not need the ‘sparseness’ conditions required for Poisson limit theorems [42, 20, 22] Possible uses for this approximation include prediction for a fitted model (i.e. calculating the intensity for a Gibbs point process model that has been fitted to point pattern data), residual analysis [3] and other diagnostics for a fitted model, approximations to maximum likelihood estimation (since the likelihood score involves the point process intensity), and stability improvements to Markov Chain Monte Carlo methods.
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