Fast, Approximate Maximum Likelihood Estimation of Log-Gaussian Cox Processes

Abstract

Log-Gaussian Cox processes (LGCPs) offer a framework for regression-style modeling of point patterns that can accommodate spatial latent effects. These latent effects can be used to account for missing predictors or other sources of clustering that could not be explained by a Poisson process. Fitting LGCP models can be challenging because the marginal likelihood does not have a closed form and it involves a high dimensional integral to account for the latent Gaussian field. We propose a novel methodology for fitting LGCP models that addresses these challenges using a combination of variational approximation and reduced rank interpolation. Additionally, we implement automatic differentiation to obtain exact gradient information, for computationally efficient optimization and to consider integral approximation using the Laplace method. We demonstrate the method’s performance through both simulations and a real data application, with promising results in terms of computational speed and accuracy compared to that of existing approaches. Supplementary material for this article is available online.

KEYWORD:

• Automatic differentiation
• Fixed rank kriging
• Integrated nested Laplace approximation
• Point patterns
• Spatial point process
• Variational approximation

Supplementary Materials

All supplemental files are contained in a single archive.

Fast LGCP Modeling—Supplementary.pdf: containing S1: Proofs and working for the variational approximation to the marginal log-likelihood for a proposed rank-reduced LGCP Model. S2: Complete and detailed results for the simulation study (Section 4). S3: Figure of complete basis function search for applied data analysis (Section 5).

Fast LGCP Code and Data.zip: Gorilla nesting dataset, R coding scripts, and saved objects to conduct the applied data analyses presented in the manuscript.

readme.txt: Description of the Supplementary Material archive, including directions for data access and use of code.

Acknowledgments

The authors would like to thank the anonymous reviewers for their contribution to the improved manuscript.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

This work was supported by the Australian Government Research Training Program (RTP) Scholarship, as well as the Australian Research Council Discovery Project scheme (DP180103543).