Logistic regression for spatial Gibbs point processes

Abstract

We propose a computationally efficient technique, based on logistic regression, for fitting Gibbs point process models to spatial point pattern data. The score of the logistic regression is an unbiased estimating function and is closely related to the pseudolikelihood score. Implementation of our technique does not require numerical quadrature, and thus avoids a source of bias inherent in other methods. For stationary processes, we prove that the parameter estimator is strongly consistent and asymptotically normal, and propose a variance estimator. We demonstrate the efficiency and practicability of the method on a real dataset and in a simulation study.

Detecting Multivariate Interactions in Spatial Point Patterns with Gibbs Models and Variable Selection

“…We shall use the recent developments that were proposed by Baddeley et al . (), which in practice conveniently lead to a logistic regression formulation.…”

“…Baddeley et al . () discussed several potential options to be used for the dummy distributions and intensities. We shall use the recommended homogeneous stratified uniform distributions and, if not otherwise stated, intensities that are four times the intensity of data.…”

“…From the available selection of optimization criteria, we picked the group lasso for logistic regression (Meier et al ., ), as a suitable penalized version of Baddeley et al . (). The group lasso is an extension of the original l 1 ‐penalization by Tibshirani () for individual coefficients and has been further extended to a mixed level penalization by Simon et al .…”

“…Starting from ideas of Baddeley et al . () we shall use a generalized regression‐based fitting approach, and so we borrow ideas from second‐generation penalized regression rather than from matrix shrinkage, even if we are estimating co‐associations rather than a mean intensity.…”

“…The Poisson regression approach is surpassed in accuracy by the logistic regression approach that was developed by Baddeley et al . (), who formulated the pseudolikelihood estimation equation as a logistic regression using auxiliary dummy point configurations. Again, very conveniently, standard statistical software can be used to fit the model.…”